'I stay with my sister until she finishes work' is a statement and therefore is acceptable but would normally be used as 'I'll be staying ....' or 'I'm staying .....'
Both are possible.
"Usually, I always stay with my sister until she finishes work."
"Right now, I'm staying with my sister until she finishes work."
The main sentence just can't describe the scene completely, that means you can't say "stay" is right or "am staying" is right.
Both can be possible, depending on the whole context.
P.S: Just my opinion!