The "old school/film" method of dealing with this was Graduated Neutral Density filters.... one, two and three stop strengths with soft, medium or hard transition serve to cover most situations. Rectangular filters in fitler holders allow you to position the transition area of the filter near the horizon in your images, so that the filter's primary effect is on the sky alone.
Problem is, most of the photos most of us take most of the time don't have a perfectly flat horizon line... so a physical filter (or even a digitally applied Grad ND) can effect other things you don't want darkened.
Today with digital you can do a much more precise job of it either by taking two photos or by double processing a single photo (best done with RAW files that have more latitude for adjustments).
Here's an extreme example where a filter would be utterly useless: Main subject indoors and shaded.... with the bright, sunlit background completely blown out. Also, moving subject makes taking multiple shots for HDR pretty much impossible. So instead I double processed the original image to make two different versions: One tweaked for exposure and color for the primary/indoor subject (left), the other adjusted to recover some of the outdoor/background detail by reducing exposure and setting a different white balance for full sunlight (center). I then used layers and masks in Photoshop to combine the "correct" portion from each into a single image, as shown on the right.
Try doing that with a filter! Even the graduated ND filters offered in some software wouldn't work.
Note: I actually ended up holding back the background a little (by increasing the transparency of the layer).... it was a bit too strong a "correction" for my tastes. And the finished example is deliberately slightly saturated or "punchy", because of the printing process that was going to be used for it.
The same technique can be done with most any single image.... It's even easier with simpler compositions. If possible to make two (or more) exposures at different settings and then combine those the result can be great. Using HDR techniques, a wider dynamic range can be "compressed" into a single, finished image and the process can be more automated in many image editing programs. To do it manually with complex compositions, such as the example above, requires a software that can work in layers & masks, such as Photoshop (but not Lightroom!)
And, yes, for deeper blue skies (and better saturation of other colors, too), the simplest solution might be a Circular Polarizing filter...
But when using wider lenses you do have to watch for uneven effects with a C-Pol. That's because it has strongest effect at 90 degrees from the light source (sun, in this case) and the filter's effect gradually tapers off at lesser or greater angle to the light source. This isn't necessarily a bad thing.... I used it deliberately for this image, to add some tonal variation to an otherwise boring, plain blue sky: