My mother is 93. My father passed 10 years go. If my mother runs out of money, are her children responsible for her care?

No, you are not responsible to take care of her, but you can expect to have social workers and nurses try to get you involved when the time comes. If you are going to be involved, you will need to have access to her assets, either as financial POA or co owner of all her accounts. You will need access to pay her bills. You will need to get this done before she becomes not competent. If you don't, you won't be able to access her money. You will also be able to apply her for Medicaid when she does run out of money.

As far as you being responsible for hands on care, absolutely not. Ideally you will want to get POA, apply her for Medicaid and place her in a long term care facility.
Then you are somewhat overseeing her care. If you don't get POA and access to her money though, none of that is possible. Social worker may at that point urge you to file for guardianship, but you don't have to do that if you don't want to (that also costs money). If you don't, at that point, she will likely be given over to the State and the state will have complete control over where she lives and is not obligated to include you in anything.

But to answer your question if you and your brothers are responsible, no you are not. If you don't get involved though, you will have no control over what becomes of mom.

What is she writing checks for? Is she still managing her own finances?

Are you her POA? For Health as well as Finances?
I certainly would not trust "strangers" (read that as non family members and or AL staff) to write checks and have access to her bank account information.
I would suggest that if you are her POA you stop this ASAP.
If you are her POA for finances you should have access to all her account information including Stocks, Bonds....
If your mother has dementia it may be to late to get her to assign POA. You or a sibling, if you want it, may have to obtain Guardianship.
If no one wants Guardianship and she is not competent then the Court will appoint a Guardian. At that point the court appointed Guardian will take over the health and financial aspect of her care.

Who is POA? That person is CURRENTLY responsible for safeguarding mom's finances IF she is medically incompetent to do so.

In the case of any aid from governmental programs such as medicaid there will be lookback of anywhere from 2 1/2 years (California) to 5 years on mom's finances. She will need her records to apply and if it is looking like she squandered and "gifted" funds she will not get coverage.

As to the family being responsible certain states have what are called "filial laws" in which family can be held responsible for helping with bills of the elder. These laws are almost NEVER implemented, and would only be in the cases of wealthy wealthy children. I cannot find anywhere a recent case where these laws were used.