Is your LW losing connection after startup, or does LW not connect at all?
If LW doesn't connect at all, try the tips here (click on link):
If LW 5 Will Not Connect
If nothing there helps then post back with your system details.
If LW loses connection after startup, in LW go tools > options > advanced > performance. Disable connnecting to DHT, disable ultrapeer capabilities, disable TLS > ok.
If you have multiple downloads in progress, pause some of the downloads or even cancel some if you feel that you can sacrifice them (if they've been sitting there for ages & not finding sources). Cancelling a download will remove it completely & you won't be able to resume a cancelled download.
If you're sharing a huge number of files, reduce the number of shared files.
Try limiting your upload & download bandwidth a bit... tools > options > transfers. Click on the Upload/Download Limits 'Settings' button to configure bandwidth settings. Click ok to save any changes.
Try updating your Java (or reinstalling, if you already have the latest update):
Java Downloads for All Operating Systems - Sun Microsystems
Download the JRE offline installer & remove any other Java versions (via Control Panel) before installing this one. Restart your computer after installing the new Java.
If none of that helps then post back with your system details.