At the Met, you'll find an entire range of dress. It varies, in some respects, based on where you're sitting. You'll find a lot of tuxedos and suits in orchestra, more casual clothes as you move further up. For the premiere of a new opera, it is often more dressy. Met attendees tend to take their opera very seriously. The kind of clothes is less important than the respect shown to other audience members and, of course, the performers, orchestra and conductor. Not applauding at the wrong times, not unwrapping candy, not talking, etc. is more important than what you're wearing.
The Met is a beautiful venue for opera and, though only built in the 70s, it is still steeped in tradition. When my wife and I go, I usually wear a nice sport jacket, nice (e.g. silk) collared shirt with no tie, slacks and dress shoes -- though it's more "dressy" than when we go to theater, I still like to be comfortable.
However, we also usually sit much farther back in the orchestra than Row C as we buy our tickets at the last minute, and also don't usually go to premieres. If it's a big event, like a premiere or the debut American performance of a famous foreign singer, I'd probably consider wearing a suit or, at least, putting on a tie.
YMMV, but one of the things that I like about opera is the formality of the experience. Live theater in New York has suffered, I think, by being promoted primarily as a tourist attraction like some kind of theme park, rock music concert, or sporting event. This has resulted, not only in extremely casual dress (which, in itself, is not necessarily a bad thing), but the casual behaviors that are concomitants, which tend to ruin the experience for audiences and performers alike. As I said, YMMV, but I think the live performing arts, whether theater, opera, dance or "classical" music, are absolutely unique from other forms of entertainment in that the audience is an active participant in the performance, and undertakes obligations beyond the passive observation that is all that is required for film and television viewing. Because the audience participates, appropriate costume and conduct should be considered.
As a side note, whenever we go to the Met, we see a sprinkling of young children, both boys and girls, some appearing to be as young as six. In every single case, though, these amazing kids have been dressed to the nines, polite, well-behaved (describing them as well-behaved really doesn't do them justice -- they comport themselves perfectly) and, most impressive of all, they've been absolutely enthralled by both the performance and the entire opera-going experience. Would that all parents instill such appreciation for the fine arts in their kids!