Is Global.asa Good?

This questions is a little one sided, I used to have the connection to the database on the site until I learned about the global.asa and since then all my sites have used it. I don't know the benefits of this file vs any other methods so I have to ask. But first let me explain.

I have a site, one of the many using the global.asa, it works 2 Microsoft access databases that run pretty much simultaneously on the site. Today, my host sends me an email telling me that they are open constantly and it's causing them problems on their shared server. So I modified the asa to close the database after every session and the host I had done so. They replied shortly after saying that a session could be open for 20 minutes before closing and they did not see this amend as helping the problem. So I'm looking for an alternate method.

The site itself is a collection database with over 1,200 user accounts, I have absolutely no idea how many of these user accounts are actually in use, but that is how many registered users there are in the database. Most of the regular users probably visit every day or two and I know that the admin (of about 8) visits daily to update the records unless there is nothing to add.

I've been given 7 days to solve this problem or they will close this particular site down. So in this case, is global.asa a good idea? If not, what method could I use to minimise the database being opened?