Connect establishes TCP/IP connection to the remote server. UseBestLogin performs authentication against remote server.
In most cases one of those methods performs SSL/TLS negotiation: Connect -or-
ConnectSSL in case of implict-SSL/TLS and UseBestLogin in case of explicit-SSL/TLS.
From technical point of view, you can store Imap or Smtp instance for later use.
However in reality it won't work. Remote server (or even one of the routers) is going to cut you off, if the connection is unused for some time.
You may try issuing 'NO OPeration' command (Noop method) at some interval, let say 10 seconds.
You may also try using Idle: https://www.limilabs.com/blog/imap-idle
Generally if you are creating web mail client consider synchronizing mail in some background thread/service and not directly in asp.net code.