You don't have to stop Crowd-connected applications as long as you have the cache enabled for the connection to Crowd. The cache is updated (synchronized) periodically, and if Crowd is down when the sync attempt happens then the sync will fail with no ill effects. If users try to log in while Crowd is down, their login attempt will fail.
The only case to be aware of is if the Crowd's behaviour changed from an old version of Crowd to a new version of Crowd. For example, Crowd 2.8 introduced new membership aggregation semantics when multiple directories sharing a user are mapped to the same application.
If you don't have the cache enabled, then users and groups will temporarily disappear (and no users will be able to log in either). Atlassian applications should be resilient to this (though it's probably not a scenario we test a lot), however third party plugins may not expect this to happen at all. Therefore, if you don't have the cache enabled for a connected application then I'd take down the connected application while upgrading Crowd.
Update: Our AMA with @Serhat Can is complete! Thank you everyone for taking the time to write your questions and participate, you are the folks that make the community enriching and fun! If...
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