The Alpinist Association Museum covers the history and culture of alpinism, with exhibits on climbing, mountain exploration, and the Austrian Alpine Club. Its setting in the Hofburg makes it easy to pair with Innsbruck's palace, church, and old town sights.
This is a specialist museum, but Innsbruck is exactly the right city for it. Visit if the surrounding mountains are part of why you came, or if you want more context before heading up to the Nordkette or deeper into Tyrol.