Treat The Walls, Not The Ceiling
The old method pushed batts up between the floor joists. It does not survive. They sag, drop out, take on water, and the crawl space stays cold and wet, so the ducting and the pipes down there stay cold and wet with it.
What works is sealing the perimeter and bringing the space inside the heated envelope. Closed cell on the foundation, carried across the band above it, over a taped sheet on the ground.
The space then sits close to the temperature and humidity of the house. The floor above stops being cold and the plumbing stops being a January emergency.
The Ground Sheet Is Not Optional
Bare earth in a crawl space releases water continuously. Insulate the walls perfectly and leave the ground open, and it will still be damp.
A proper sheet goes down first, lapped, taped at every seam and carried up the wall far enough that the foam ties into it. That is what makes the assembly continuous rather than merely thorough.
We do not leave this out, and a quote that does is worth asking about.
Access Drives The Price
Crawl spaces are awkward in ways that vary enormously. Some have a hatch you can barely get a shoulder through. Some have services running the full length. Some have under two feet of clearance.
All of that changes how long the work takes and therefore what it costs. We look at the access before quoting rather than finding out on the morning, so the figure you are given is the figure you pay.
Full protective equipment and supplied air are standard for us in a confined space. That is not a preference, and it is worth asking any contractor to confirm.
About Those Vents
Vents went into older crawl spaces on the theory that outdoor air dries a damp space out. Beside the Atlantic that theory inverts across much of the calendar: warm humid air entering somewhere cool condenses on the first cold surface it touches.
Closing the vents and sealing the space is the current approach, and everything above depends on it.