In American English there isn’t an argument on this one – it’s always an inquiring mind but in British English the jury is still out and it seems that either spelling goes.
Why the confusion?
Well, ENquire is the Old French and Middle English form and INquire is the Latinized form – it seems both had pretty current usage since written English became established.
Some people feel that there is a difference in meaning and that ‘enquire’ means to ‘ask a question while ‘inquire’ means to investigate. If you’re in the USA, as I say, this isn’t an issue but for the rest of us it seems that you can still choose between those who follow Chaucer (enquyrid) and those who follow Shakespeare (enquire).