Frontier is as safe as any other carrier. Statistically, they're slightly less reliable. They have low customer satisfaction ratings but I put that down mostly to customers not understanding what they're getting into w/r/t extra fees and unbundled service / comfort options.
It's just a different product from the full-service network carriers. Some people don't get that, and flip out when they find their $49 Frontier ticket doesn't buy them the same network resiliency, service recovery, etc. they would have on a $200 fare on Delta or Southwest. You don't go to Jack in the Box, though, and expect Ruth's Chris service levels. Live and learn.
Fine to fly Frontier with two caveats:
1. Watch the ancillary fees for carry-on bags, seat assignments, etc. -- that "bargain fare" may turn out to be less of a deal than you think.
2. Be prepared to roll with the punches when weather hits or planes break -- sometimes for a day or more. I've flown my kid to and from college on Frontier with no problem, plenty of slack in our schedule. But I wouldn't fly F9 myself on business, or in any case where I had to be somewhere at a specific time.