Does having one (small) joint account with Mom risk my other personal bank accounts?

Mom put my name on a $20k stock market account so that I could trade stocks about 5 years ago. It's a joint account in our names. I always listed it on her taxes (not mine). No money has ever been withdrawn by either of us. Because we have this one account as joint, can senior housing require me to pay from my other personal accounts (that do not have her name on them)? [Note: Mom's in the hospital now - not yet in Senior Housing - but I think she should consider it.]