How can I clean cobwebs and a musty smell out of an old basement?

A broom can get the cobwebs. I normally use the wand on my vacuum though. For the smell, there's a product called damp-rid that can be used to soak up the smell.

About the cobwebs, yes, use a shop vac or a vacuum with a long hose to reach. I couldn't understand the question about the cobwebs

As for the smell:

Get a dehumidifier.

keep the dehumidifier on constantly. You will see the smell will disappear. If you are dealing with a moldy smell, then that dehumidifier is the key! It may take months for the smell to dissipate but it should work. AND keeping water out!! You might have to fix any issues with your gutters and adding some drainage tiles in your yard underground to carry water away from your house so you don't get any flood waters coming in which will give all the musty moldy smells.

I would use a broom to remove old cobwebs from the rafters and dehydrator to minimize the dampness. You may need more than one and leave them running until the dampness edge is taken off then leave at least one running all the time. You might try painting the walls with paint made for damp basements if it is cement or cinder block walls too. If there is paneling or any wood on the outside walls, it may need removed if it has been very damp for a long time. Good luck and hope this helps.

The shop vac hose should reach the rafters with the wand attachment. I agree on the dehumidifier. Walmart sells them at one of the best prices. Run it daily and make sure you empty it as soon as it is full.

Here’s some effective tips for the musty smell: https://www.frugalfamilytimes.com/2019/04/how-to-get-rid-of-musty-smell.html?m=1