There are a lot of problems shooting basketball and most point-n-shoot cameras will simply not do well.
You need quick autofocus, a bright lens, relatively fast shutter speeds, usable higher ISOs and, ideally, some means of dealing with the fluctuating lighting in most gymnasiums.
I'm going to focus on Canon gear because that's the system I know best. There may be some other manufacturers' cameras with similar capabilities.
Among Canon I'd recommend a used or refurbished Rebel T7i (800D outside N. America), EOS 77D or EOS 80D... Or a current Rebel T8i or EOS 90D.
Here is why:
All these cameras have Canon's 45-point AF system that's fairly fast and capable. All 45 of those AF points are a higher performance "dual axis/cross" type. The 80D and 90D have the best performing version of this AF system. You can use a single point (most accurate but most difficult), small groups of points (useful for fast action, though less reliable), or enable all 45 points (least reliable). While there are other Canon cameras with even better AF for fast basketball action (7D Mark II, 1DX Mark III, R6, R5, R3), the above models are reasonably affordable AND have a lot of automation features to make them easier to use. The 80D and 90D also have a "true" pentaprism that makes for a little bit bigger and brighter optical viewfinder than the T7i, 77D or T8i, all of which use a "penta-mirror" to save weight and cost.
The T7i, 77D and T8i have a fastest shutter speed of 1/4000... while the 80D or 90D both feature an even faster top speed of 1/8000. Fast shutter speeds are necessary to freeze sports action. You will find many point-n-shoots max out at 1/2000 shutter speed. That may be sufficient, but there will be times when even faster would be better.
All these cameras use 24MP APS-C size sensors, except for the 90D which uses an even higher resolution 32.5MP APS-C sensor (the highest resolution of any camera with this size sensor). 24MP is more than enough for sports photography and APS-C sensors are considerably larger than what is used in most point-n-shoot cameras. This allows higher ISOs to be used before digital noise becomes a problem (still bigger "full frame" sensors are even better in this respect, but make for bigger, heavier and more expensive cameras and lenses).
Canon T7i, 77D and T8i all can shoot continuously at 6 frames per second. 80D can do 7 fps and 90D is even faster at 10 fps (faster with locked AF, but that's not something you want to do when shooting sports). Continuous shooting rates are important to catch fast action... up to a point. They can cause you to take an awful lot of images that will need to be downloaded, sorted and worked with later. But a short half second burst of 3, 4 or 5 shots can capture the "peak moment" of action. This goes hand in hand with "shutter lag", which is the time between when the photography presses the release button to take the shot and when the shutter will actually release. There can be quite a lag with a point-n-shoot, which can cause photographers to miss a lot of fast action shots. DSLRs and mirrorless cameras can do better, but there's some variation among them, too. For example, the Canon 80D's shutter lag is a fast 0.060 second. The less advanced T7i is good, but is about 30% slower with 0.079 second shutter lag. Or a faster 0.052 second is possible with the 7D Mark II or even 0.038 sec with the most advanced 1DX Mark III (Note: All these are "optimized" shutter lag, tested with autofocus turned off. This is used for camera to camera comparisons. When AF metering are active, as they usually will be, there is always some additional delay.)
Yet another feature that's found on all the above Canon cameras is "flicker reduction". This was first introduced on the Canon 7D Mark II that I use and it's a real game changer when shooting "under the lights". The type of lighting used in many sports venues actually cycles on and off very rapidly... 60 times per second. We don't see it with our eyes... but our cameras sure do! In cameras without flicker reduction it causes A LOT of your images to be incorrectly exposed.... badly underexposed. Previously the only solution was to use a 1/30 or slower shutter speed... but this doesn't work for sports photography. You need faster shutter speeds to freeze the movement of the players, balls, bats and such. When shooting sports the best we could do was take lots and lots of extra shots, knowing that around half of them would be badly underexposed and a lot would be unusable.
Flicker reduction largely solves this. It detects the light cycling and times each shutter release to coincide with peak output. Using it I now see very few poorly exposed images, even when shooting familiar locations where the lighting gave me trouble before. Canon has put Anti-Flicker into almost all their cameras since 2014 when the 7D Mark II was introduced (the more entry-level T7, SL2/SL3 and, other than the M6 Mark II, M-series mirrorless don't have it). You can see more about this feature here:
https://www.the-digital-picture.com/Canon-Cameras/Anti-Flicker-Mode.aspx I know Nikon has a similar feature in a few of their cameras too.... Don't know about Sony or other manufacturers.
The lens for this purpose is going to be more challenging. An ideal lens has high performance autofocus and a large aperture to be usable with a fast shutter speed in limited lighting conditions. A zoom can be desirable for the versatility, but prime lenses can offer a larger aperture in smaller size, lighter weight and lower cost. The "kit" lenses offered with all the above cameras can be good, but are NOT large aperture and many don't feature the fastest type of focusing motors.
The most ideal zoom lenses available for basketball that can fit the above Canon DSLRs:
- Sigma 50-100mm f/1.8 DC HSM... great focal length range for basketball... over a stop larger aperture than other zooms... no image stabilization... pretty big & pretty heavy and $1000.
- Canon EF 70-200mm f/2.8 IS USM III.... superb lens, pro build... slightly longer than usual needs for basketball... "only" f/2.8.... large, heavy and over $2000 (Sigma & Tamron versions cost a lot less).
- Tamron SP 24-70mm f/2.8 Di VC USD G2.... good lens... slightly short for basketball... "only" f/2.8... moderately heavy, $1200 (less than the Canon & Sigma versions).
- Canon EF-S 17-55mm f/2.8 IS USM... great lens... most compact and lightest weight... "only" f/2.8... pretty short for basketball... most affordable under $900 new, widely avail. used for less.
All the above have their manufacturer's fastest type of auto focusing motor: Canon USM, Sigma HSM and Tamron USD.
Prime lenses I would use to shoot basketball are 50mm f/1.4, 85mm f/1.8 and 135mm f/2. While less versatile, these are one to two stops brighter than f/2.8 aperture zooms. They also are typically a lot smaller, lighter and less expensive. I would also consider using 35mm and 100mm lenses. And I'd look for those same fast focusing motors in these primes.
Ultimately lenses for sports ain't cheap. If willing to consider refurbished (direct from Canon USA, same warranty and little difference from new, except for the plain brown box). When they are in stock, refurbished copies of most of the above recommended models will cost less than most of the above lenses:
https://shop.usa.canon.com/shop/en/catalog/cameras/refurbished-eos-interchangeable-lens-cameras#facet:-81026611110012132791101081213275105116115,-810369798332555568,-810369798332564868,-8103697983328210198101108328455105&productBeginIndex:0&orderBy:&pageView:grid&pageSize:& For used cameras, check out B&H Photo, Adorama, KEH.com, MPB.com, usedphotopro.com (Roberts Camera), and others.