If you are remodeling plan on a new sink. If you have to, cut other things from the budget.
Refinishing will cost more than a new sink.
Applying a sealer to the cracks will look like an amateurish bandage on an old item in a new remodel. If it should fail you risk water damage to the cabinet, floor and possibly more. I don't think this approach is worth the risk.
To help fit a sink in your budget, try shopping at Habitat-for-Humanity stores or other salvage or discount home supply stores. A new or slightly used vanity may need a coat of paint to match your decor, but the price can be surprising.
(Last week I found a 48" new vanity with counter and built-in sink for $115. solid wood with dovetail drawers!)