I have about 1500 miles and my strength and fitness are improving. I notice that when I spin hard, there is a tendency to let the bike sway to the side. What is the proper technique, let the bike sway or try to keep the top tube still.

Rod

If by "spin hard" you mean putting out power at high cadence, I'd say that swaying could be indicative of an imbalance in your left/right power output/leg strength, and this would be something you'd want to work on remedying. It might also suggest your cadence is not as high as you feel it is.

If by "spin hard" you mean to grinding out power at relatively low rpm, then it's natural to have this swaying, and it's nothing to be concerned with.

^^^+1. It's situationally dependent. Seated and spinning at 100 rpm there is usually very little if any side to side motion. Standing at 60 rpm there is a lot as body weight is shifted from one pedal to the other. There is a continuum between those two extremes. The goal is to minimize extraneous energy expenditure, which can come from fighting the natural tendency/need to sway, or on the other hand, swaying needlessly/excessively. It's something you gain a feel for with experience.

I can stand and hammer, or spin, and not let the top tube sway much, or let it sway. As long as I steer straight, there is not a big difference. Now if you also weave, that seems to be bad. Easier, but bad.

Rod

First there is a mountain...then there isn't.

I would say unless you are a professional racer, dont worry about it. Just ride any way that feels right.

I've seen video of racers that did a lot of side-flopping of the bike when standing and riding uphill. So it must not be too terrible.
I found with a tandem, you can't afford to do that when a second person is seated on the bike..flop it to one side and it doesn't flop back!
So you learn to stand and pedal without flopping the bike around.
If all else fails, try it both ways and see what works best.

I think with technique at least with me I like to go at a higher gear so I can keep the rpms high