Baseboard was not removed when new drywall installed. How to fix
An alternative that may work for you, depending on the type and height of the existing baseboard, is to replace it with a taller one. Probably need a couple inches above the existing height to make it cover the bottom of the drywall, especially where it has started cracking. If your existing baseboard is 2" - 3" tall then remove it and install ~ 4" - 5" to replace it. If your existing baseboard is already > 4" tall then this may not work so well. Cutting and painting baseboards is a lot of work - but a lot less than patching and painting all that drywall.

If you are going to stain the new baseboards, definitely do that before installation. If you are going to paint them (typically either a gloss or semi-gloss variety of the same color as on the walls or sometimes either plain white or a contrasting color - totally subjective), you can paint them before installation (measure, cut, paint, install) and then touch-up after installation, or you can paint after installation.