Soccer needs to be a fall sport or a spring sport. We need to bring on the marching bands, the cheerleaders and the crowds to the games. There needs to be more pom pom.
I'm not going to read all the replies, but since this completely veered off course I'll do my small part to swing it back slightly.
I think there are a few challenges. one is field space. Many schools just don't have a lot of field space to allow teams to practice or play games. so you have to squeeze it in when you can, primarily after football, before lacrosse, before baseball. My kid's team would walk a block to a nearby elementary school on some days just so all 3 teams could have field space.
For crowd sizes, yes the 'event' of football doesn't translate well to soccer, and doesn't have a tradition of pep bands like football does. Maybe you could get a band director to show up? but they're burnt from marching season by the time football is over and they want a break too (source: 1 kid was in band, the other played soccer). A large chunk of HS football crowd is family and friends. A soccer roster is just so much smaller, so even when the parents show up it's 30 maybe 40 people per side. Football is what 40 person roster?
For my kid's high school soccer the problem was really kickoff time. even varsity many times would play at 3pm, 4pm. One game per season against the cross town rival would be a night game.
But as you said, it would go a long way if the boys would attend the girl's games, and vice versa. Same goes for any activity. I remember one year at my kid's school the football coach gave off-season football players extra credit for attending an orchestra concert. Band supports them all season, it was only fair he said that they show up for one show.
I think a small part of it, at least in southern california, we are not generally a culture that supports lower league teams. USL, NISA, ECHL, WCHL, whatever else you want to name. Attendance is low at those events. High School soccer viewing is not a much different mindset.
Could it change? yes. but it would take a lot of lift from activity directors, ADs, opposite-gender coaches, the entire student body. Games are not always held in stadiums so that presents it's own problems.
As participation in football wanes, it is in teams' and schools' best interest to find other activities to raise student body engagement.