Which version of LW are you using?
When you see the disconnected message, do your current downloads ALWAYS stop?

I'm using LW 2.4.1 Pro now and am often "disconnected" during downloads.
Sometimes LW reconnects w/o my doing anything...although my download
speed increases when I'm "disconnected" and current download(s) do continue
(unless my host goes off-line) to completion.
However, once the downloads are complete, if I'm not reconnected by LW,
I too have to shut LW down and restart...the "connect" button doesn't work.

If your problem is with this "connect" botton in the LW>file menu,
I can tell you that the "connect " button hasn't worked for me
with ANY of the LW 2.x.x versions, free or Pro.

LW 1.7x is the ONLY version with a WORKING "reconnect" option and I use LW 1.7 when
trying to "chase" or "daisychain" downloads from a specific host.

Another disconnect reason (for me) is the internet access through AOL...
If I am "disconnected" from LW, AND the downloads have stopped,
it usually means that under my LW window is a message from AOL:

"The AOL host computer is not responding...please sign on again"

This usually means a force quit from everything and a Mac>AOL>LW restart.
LW 2.4.1 Pro (usually) does resume (in-progress) downloads
when up and running again...from the achieved %point of the disconnect.

All LW 2.x.x prior to LW 2.4.1 did to me as you have said with "resumed downloads"...
restarted from 0%