Yes. Anytime you give out your personal information to a third-party, there is always a risk that the information will be stolen or leaked. Robinhood has a good track record of protecting users’ Social Security numbers, though, so providing yours is as safe as giving it to most reputable financial apps.
Unfortunately, U.S. FinCEN requires you to provide your Social Security number to your stock brokerage for identity-verification purposes. It is very difficult to avoid the requirement if you are a U.S. resident or citizen.