There are a lot of dropped ceiling styles I believe. When I was a kid there was only the 2x4 foot panels that were used everywhere. I would do what DH wants as he is the one who is going to be needing to access the area.

If it's unfinished, can't you just NOT put anything there that will need to be accessed?

A dropped ceiling doesn't have to look bad, but I sure wouldn't install one unless absolutely necessary. Do you trust your GC to tell you what he really thinks or is trying to remain neutral?

springplanter:

Yes, you may need access once every decade or so. So cut the drywall and patch it. Nothing screams "basement bathroom" like a dropped ceiling.

What do you think you need to have access to? If there are electrical connections, they need to be in a junction box, which you can put on the finished side of the ceiling. But since you have access now, you would be better off eliminating those kinds of connections altogether. As for plumbing access, you only need access to shutoff valves. If they're in the joist bay, you should move them to where they will be readily accessible. Look at it this way: are you also planning on making all the finished walls in the house into full-size access panels? Plumbing runs up the walls, and you don't allow for it there, so why would you in the ceiling? Do you have dropped ceilings in your living space upstairs? There's usually plumbing in the joist space between floors wherever there are plumbing fixtures above. I would not do a dropped ceiling in this instance, especially since you don't like it.