You should always close the connection once you're done with it. Otherwise it will takes eat up your server's memory. Best thing to do is to open the connection, do what you have to do and then close the connection.

If you have only one database that you use, you shouln't have to use more than one connection though. You will have to use more than one command/recordset objects if you have more than one request for sure,but then again you should always close them as soon as you're done with them, especialy the recordsets.