-Hold them to high standards in everything you do. Insist that they do it right before you move on. I've made entire classes leave the band hall, line up in the hallway, and re-enter the room (yes, in high school). If they don't play concert F together, we'll spend the entire period on concert F. You may have a couple of class periods of drudgery. You very well may lose kids over it. Eventually they'll either get with the program or find a different program.
-Build up the individual musicians. Someone below suggested ditching all-region - I say just the opposite. Work that all-region music in sectionals. Make it a grade (in small chunks at slow tempo). You'll have some kids who start to get it, and more importantly start to realize that they CAN do it. They'll start to see their friends having success at all-region. Success breeds success.
-Put some individual responsibility on the kids. Chamber music is great for this. A lot of kids don't care about their grade, but they don't want to let their friends down. Put them in a quartet, one on a part. If you have the right numbers, you can have multiple quartets learn the same piece, but they perform one on a part. Make sure they know there will be a public performance in front of the class, in addition to the contest.
-When you DO have success, make it a BIG deal. If you have a kid make Region, put it on the school's Facebook, morning announcements, band newsletter, everything. Chamber group qualifies for State? Pull out all the stops. Send a press release to the local paper (with photos!). Recognize them at a school board meeting. Make sure the kids get recognized at the all-school awards assembly at the end of the year. Get a perpetual plaque, start adding kids' names. Look around at the successful organizations on campus - sports or whatever - see what they're doing and copy it. I know my school still has photos around of the state championship basketball team from the 90s, no reason the band can't do the same. Celebrate the successes, other kids will see that.
-Make Region, Solo and Ensemble, especially any state trips a big deal. We go out to eat at a nice-ish restaurant during the Region clinic. For most of my kids, even Applebees is a step up from their usual. The State Solo and Ensemble trip is a big reward for those who qualify, with a night in a hotel and a trip to the water park after the performances.
-Stop teaching by rote. Teach the skills needed to read. Do sight reading of some sort every day. Even if it's just a simple rhythm. Get a rhythm book and work through it - 100 rhythmic rest patterns, Winning Rhythms, Louis Belson books, something that progresses through the year. Drill scales. If the kids can read rhythms and play scales, they'll be in good shape for sight reading. This may take a semester of drilling fundamentals before they're ready to move into any real music, but it's worth it.
-Build up some student leaders. Start with the ones who care. Give them some real responsibility. Keep an eye out for next year's student leaders and start building that now. Maybe they're freshmen - age is less important than attitude.
-Grab those kids who do care, and find a way to make it fun to care. Student leadership can be a big deal. Find a way to feature them on a performance. Get them out of class for a day and go recruit at the middle schools. Publicly celebrate their successes.
-Make it a big deal to make the top ensemble. They play harder music, but do they get to do anything extra? Take an extra day out of class for another concert festival? Middle school recruiting? Take small groups around town to play Christmas music - retirement home, fire station, town hall, wherever. These kids can afford to skip school for a day and go play some music. Take them out to lunch somewhere while you're at it. Make it fun and exciting to be in the top group, something kids in the lower bands can look forward to and strive toward.
-Make sure the lower ensembles are preparing kids to join the top group. Whatever your audition requirements are at the end of the year, you should be teaching that all year. Whatever you expect of your top group in August, your second group should be able to do by May. Make the top group attainable.
-Drop the dead weight. Prune the tree. Cut out the cancer. Pick your metaphor, but get rid of the ones who are really toxic. Put out some very specific, challenging, but achievable standards. If they kids aren't meeting the standards, it should be reflected in their grade, and now they need to find a different elective. Find out when your school will allow schedule changes and make sure you're out ahead of it - have your documentation ready, make sure you've been contacting parents, if there's any pushback you can present a very strong case that you gave the kid every opportunity to meet the standards and they have failed. Once you've done this for a year or two and the kids realize you're serious, most of the ones who don't care will show themselves the door. I remember one of the "perspectives of the honor band directors" sessions at TMEA where one of them described the 10-80-10 rule. 10% of your kids are the awesome leaders in the band, 80% are just there, 10% need to go. I'm not that specific on my numbers, but I definitely look for quality over quantity. To be clear, this is much more about attitude and work ethic than innate talent.
-You need to have all of the directors on your campus on the same page here to be really successful. I'm guessing you're an assistant, if your head director isn't ready to put in the effort then you're going to struggle. Apply this to whatever part of the band you're in charge of. It's much easier if everyone is on board, and SO much easier if it starts in middle school.