What a sad situation and how difficult for you.
It can be a shock when approached like this. Many kinship carers become carers/parents overnight, for one reason or another, and it is nearly always a complete shock. You neither have to nor don't have to step forward for this child, but you should have the full information before you decide.
I recommend you contact the charities Kinship or the Family Rights Group tomorrow. They answer the kinds of questions you have daily.
I'll focus on the financial side as that is what has been mentioned most, although there are also many other considerations. First though, rest assured that for a baby it is more likely than not that a permanent, loving home will be able to be found should they not be able to stay with their birth parents or a kinship carer.
Are you in England? If not you might need to ignore the rest. If so then as a kinship carer if you are a foster carer you should get the same allowances and have the same support as any other foster carer through the local authority. LAs are not supposed to discriminate against kinship foster carers compared to other ones. You should be able to get details from your LA, but you can also look at the Fostering Network for further information on allowances etc. You are also supposed to be eligible for the same "skills allowances" as foster carers. This means that once you have undergone further training, you can get a higher allowance. These differ between LAs. Fostering allowances also have a very high tax-free allowance, on top of your normal one. I believe they are in the main discounted for benefits, but there will be more information on the Fostering Network's website about this.
A problem arises if the LA tries to place a child with you "voluntarily". This happens a lot and means they will not pay you as a foster carer. You must make sure it is as a foster carer, and certainly at least initially, even if you later go on to have a Special Guardianship Order or adopt (if appropriate). It's important the child is "looked after" (ie in care) at some point.
A fostering allowance is non-means tested and lasts as long as the child is "looked after". If in the future you go for a Special Guardianship Order, this will have financial implications. LAs are supposed to offer financial support - a key difference is that it is means tested. For SGO allowances (different to the allowances for testamental guardians mentioned above), many LAs - but not all - use a means test also used for Adoption Allowances. Therefore the more you earn, the less you get. The total is often, not always, tied to the LA's fostering allowance. Eg in my LA, the full amount is tied to the basic fostering allowance (ie. the basic allowance not including any skills payments a foster carer might have on top due to training and experience) minus child benefit (as you can't claim this for a foster child but can for a child under an SGO). However, this is "discretionary" and LAs can and do change the amounts and the means test, although I do know of some who have negotiated a fixed amount until 18, rather than means tested until 18. However, there are other advantages to an SGO.
Other points mentioned -
Kinship carers can only foster with local authorities, not other fostering providers, so can only have the fostering allowance offered by their particular LA.
Also, sometimes, not always, requirements such as whether an own room is required are different for kinship carers. However, it seems unlikely that a 12 yr old and baby would be allowed to share.
I hope this helps. I would have a look at these websites and maybe call them too (I can't do links at the moment sorry):
Kinship
Family Rights Group
Fostering Network