What do you mean by "download from the web"? What would be wrong with this happening, anyway?

If the account uses POP then an email client will try to "download" all it is offered and tell the server to delete it. Since it can see only the Inbox in a POP-connected account, you could move material that you don't want or need to see in Thunderbird to a separate folder on the webmail site.

IMAP, on the other hand, will attempt to show anything and everything on the server (though you can intervene and unsubscribe from folders you don't want or need in Thunderbird.) But IMAP doesn't "download", in the sense that it makes local copies in Thunderbird but leaves the originals on the server. But be aware that anything that you can see in both Thunderbird and on the server will vanish from both, if deleted, UNLESS you make a local copy, perhaps in Thunderbird's Local Folders account.

What are you scared of? Losing old precious messages (copy them to a separate folder in webmail, make sure it isn't subscribed) or running out of space (in your own computer? or on the server? In both cases archiving to a separate storage medium may be necessary.)

Thanks very much for the reply. Yes the account does use POP. Thunderbird was my email client and my emails were stored on my box. But the box had to go into storage for a year (I was homeless) and I viewed my emails on the Web. When the box came out of storage I had a huge number of emails sitting on the Web site. I was able to open the locally stored emails using Thunderbird and I didn't click on 'Get mail'. The box died a week ago and I need to install Thunderbird on the new one. But of course it asks me for my account details. I want to delete most of the emails on the server before asking Thunderbird to fetch but I need to be able to read my locally stored emails. You have given me the answer - move the mail on the server into another folder. Thanks very much.