You can disable the web UI notifications altogether under Admin → Settings → Metrics & Profiling → Usage Statistics and turn off the option for enable version check. The downside is you won’t see it anymore in the web UI, and unless you see the forum posts here when new updates are made, then you might not necessarily know. I tend to update my servers regularly anyway, so would see the package appear when updates are released anyhow without subscribing to notifications. I guess it’s just personal preference on how you administer servers.
There doesn’t seem to be any other way than this right now, eg: limit it to email notification when a new update is available or some other method. It would be good though if Gitlab expanded the abilities for what kind of notification, GUI or email or whatever.