Sealed bottom bracket bearings exist, but are uncommon. The grease is
typically viscous enough not to flow out of the bearing. Just after
assembly, you'll get a little squeeze-out, which can easily be wiped
off. It should be fine after that. I only use Phil Wood waterproof
grease, which is supposed to be better for this sort of thing.
Of more concern is the possibility of contaminants entering the bottom
bracket from other places in the frame. Lots of older and new cheap
bikes have no sleeve in the bottom bracket, so anything that finds its
way into the bike frame can drop into the bottom bracket. If you have
internal cables, unused braze-ons, or an opening in the top of the
seat tube, water and dirt can easily get in. I take an aluminum 12oz
soda can and cut a metal strip wide enough to just fit between the
outer bearing races. Curl it tightly enough to slip between the races
and allow it to spring outward against the inside of the bottom
bracket. Works like a charm.