It might be worth checking out the subfloor if you can get access to it in case that is what is not holding up to weight instead of the joists.
Otherwise, I know in your comment you stated that you don't have the joists exposed, unfortunately, I don't know of a way to resolve the issue without exposing them as anything you tried to do at the floor level would just be adding weight.
If you don't feel your joists are up to snuff, you can reinforce them by laminating them.
You would get additional boards of appropriate size to fit snugly next to the joists, then use an adhesive and screw the boards to the joists.
Make sure you are getting the additional wood over your load bearing walls, otherwise again you are just adding weight to the situation.
Remember though, even if you are going to make it a DIY project, structural work is a big deal and you should at least consult a contractor/architect and make sure you get proper permits and inspections.
You can save yourself the money on the labor, but don't skimp on the planning.
There are several issues here to deal with that I see.
First, you need to find an appropriate location for this aperture. It is best if the location is one that you can get to easily, but out of the way - since a hole in the floor might be unsightly or something to trip over. It also needs to be accessible from below to get in and out. I would suggest a back hallway or back bedroom as being good.
Second, you need to worry about which joists are involved. Can you get through a hole between the joists in your house? If you needed to cut through a supporting beam, this can have very serious and negative consequences on the structure of your house. Do NOT do that without knowing what and where you are cutting, and providing the necessary support for anything that is cut.
You will need to carefully mark out the cut lines. Careful measurements will be important. You may choose to make a tiny hole or two with a small drill bit to figure exactly where you will cut relative to any joists.
Last, the actual work. Cutting through the floorboards, once they are carefully marked, requires no more than use of a handheld circular saw or a reciprocating saw. There are several other types of saw that could do such a cut on the market, rotory "zip" tools for example, even a hand held manual saw if your skills were adequate. If you needed to cut a beam, then you will need to provide new support points for that joist on either side of the cut that are FULLY capable of carrying the load. You may even need to reinforce the floor boards themselves as they were cut on either side too.
To close it all up, take the flooring that you cut out, and create a door from the flooring itself that matches the hole you just cut out. Recognize that this will be a trip point, so do not put a big, bulky handle on it. A simple, low piece of moulding around the perimeter may be sufficient to hide the cuts and support the "door". Even that is a trip point, so be careful here.
All in all, this MAY be something that is best given to a reputable contractor to solve for you if you don't know how to solve the problems that will arise. The cost would depend upon whether you can find a location that does not require the cutting of any beams.
Best Answer
It sounds like the joists aren't properly fixed to the walls or they aren't big enough for the task at hand - keeping your floors up.
I'd get another builder in to take a look and quote for remedial work and then try and get the first builder to either sort out the problem or pay for someone else to do the work.