Can one leave one’s non-Jewish maid at home without anyone else home as supervision? Or is there a problem of her possibly cooking and treifing up your kitchen utensils? Are there any heteirim for this?
She can be left at home on her own, and should be told not to cook anything (with the home’s utensils).
An interesting comparison to this question is the matter of wine left at home.
For bottles of wine left at home, if the maid does not know that the owner is going out for a lengthy period of time, the wine is permitted.
If she knew that the owner will not be returning home for a while, and she is from a people who practice idolatry, the wine is prohibited.
If she is from a people who do not practice idolatry, the wine becomes prohibited only if she was left alone with the wine for a considerable period of time.
Concerning bishul akum, the matter will appear to be more lenient. The very question of beliyos of bishul akum is disputed by rishonim, the Beis Yosef mentioning two opinions (113:16), citing first the stringent opinion of the Rashba, and then the lenient opinion of the Rosh, Ra’ah, and others.
Although we are stringent for this halachah, if the maid is told that she is not allowed to cook at home, the matter will surely be no more than a safek, and therefore one can be lenient. Moreover, cooking can usually be discovered, and one can therefore assume that she won’t cook, for fear of discovery. The special concern of wine, which is a matter of idolatry, does not of course apply.
Sources: For wine, see Shulchan Aruch and Shach, Yoreh De’ah, 128:1; Bein Yisrael Lenochri, pp. 349-151; see also Shulchan Aruch, Yoreh De’ah 123:25; Taz 24; Shach 74; Shulchan Aruch,Yoreh De’ah 124:24.
See also here for further material on this subject.