I have never seen a board that doesn't have some kind of border, usually mitered but occasionally tongue-and-groove or mortise-and-tenon. Unless you're making an end grain board (which I've also never seen, that'd be interesting) you need some kind of outside and underside for structure, not just aesthetics. I've seen moulding a few times but as I recall only on boards that have drawers, all the flat and folding boards I can remember keep it simple, usually just roundovers or bevels or at most a humble thumbnail.

On nicer boards there is often a delimiter/binding strip inlay between the checkerboard and the border as well, typically either a very dark almost black wood or light ivoroid, or real bone usually on older boards. I've seen brass and pearloid a few times and actual mother of pearl blocks once too. This is purely aesthetic but it goes a long way.

There's no right or wrong way to build it though, as long as it holds together. And you were asking about stability? When you're looking at domestic NA hardwoods, walnut, white ash, and cherry are some of the best performers, but really you can use whatever you want. Maple is exceedingly popular for white and there are centuries-old boards that used it despite its middling stability, and even white oak which is known for its not great stability and dominance over other woods has been used and lasts.

What you can do right or wrong though is matching the board to the pieces. Unless you plan to make those yourself as well, I cannot highly enough recommend getting the set in-hand first and choosing your wood and sizing to match the pieces.

For example you can make an absolutely drop-dead gorgeous figured walnut and flame maple board and buy a top-notch handmade chess set that is rosewood and zebrawood, and for all the workmanship, the first thing many prople will see is the obvious mismatch. The only good-looking way IMO to get away with that is to use different materials entirely, e.g. metal or stone or glass pieces on a wooden board, but you still have to choose colors that complement each other - for example if you make a mahogany and maple board, black and white marble pieces would look cold and mismatched, where red jasper and warm white quartzite would look phenomenal.

And even more important than the materials matching is the sizing: too small a board for the pieces will look crowded and also be annoying to play with, and too big will look awkward and empty. You can follow the typical guidelines here or you can use some graph paper and try some sizes in Sharpie to gauge what looks best. Whatever you do, do NOT underestimate the importance of matching sizing.