It's the splasher that needs to change not the sink so get the sink you prefer. I'm planning to add a 6" backsplash in our bath remodel for this reason, but need to have something handy for the splasher to wipe up after using the sink. May be a losing battle....lol.

With little space, an all-one-piece porcelain sink and countertop will minimize drips. Pick one with a lip at the edge to keep water from trailing down the front. Your faucet choice matters a lot, too. Use a low simple faucet that doesn't stick out far towards the bowl. Some people keep an extra hand towel in the bathroom just for mopping up.

How old is the splasher? If a child teach now to wipe up the mess if an adult IMO same lesson.