Hi Brendan Frank,
Thanks for using the Apple Support Communities. From your post I gathered that your email account in the Mail app goes offline and you also experience issues with Safari not responding, but only on the MacBook Pro not the iMac. I like to check my email on multiple devices so I can understand your concern. I will be happy to help you with this. I found this support article that goes over some information:
If you can’t send or receive email on your Mac
Particularly this section:
If Mail says that your account appears to be offline
Mail might say that your account appears to be offline due to a network problem.
If you're connected to the Internet but your account appears to be offline, your email settings might no longer be correct. To view your current settings in Mail, choose Preferences from the Mail menu, click Accounts, then select your email account.
OS X automatically uses the correct account settings for many email providers. You can use Mail Settings Lookup to view those settings, or get the settings directly from your email provider. Then compare them to the settings you see in Mail preferences.
If your MacBook Pro hasn't been restarted recently, I'd recommend a restart.
If Safari is slow, stops responding, quits unexpectedly, or has other issues, this article goes over some information for the Safari issue you're experiencing as well.
You can also test out both of these issues in safe mode to see if they occur there. You will first need to add the email account in the Mail app.
Try safe mode if your Mac doesn‘t finish starting up
To get back into your main user account after testing out safe mode, just simply restart the Mac. To check to see if these issues are user specific or system wide, you can always check this in another user account.
How to test an issue in another user account on your Mac
Have a great day!
