Musty smell in a basement is one of those things that is difficult to assess remotely. Some HVAC technicians don’t like to investigate such problems because it might be time consuming. Often, because of our senses, what may appear as unpleasant / musty smell for some of us, it will be perfectly normal for others.
There are many factors that might be responsible for what you’re describing but without knowing specifics of your HVAC system installation, house drainage, foundation type and condition, plumbing system materials and their condition, it is virtually impossible to pinpoint the source of your problem.
If you didn’t already, please take a look this article on my website - http://www.checkthishouse.com/3258/musty-smell-from-heating-system-air-ducts-forced-air-heat.html
Some of the things to check / tips:
If you furnace stands on the basement’s floor and / or if you have any air ducts under the concrete slab read that - http://www.checkthishouse.com/14/air-return-in-forced-air-heating-ac-systems.html
Furnace blower / air return compartment should be checked for moisture / mold contamination.
If you have air return ports in your basement area you can install dehumidifiers to lower humidity levels
Humidifier units are huge sources of mold and moisture. They require servicing / maintenance on regular basis when used. How often depends on its type – check with the manufacturer. Many of the humidifiers waste tons of water and are difficult to control so it’s a good idea to have an independent humidity sensor in your house to verify the output. Read it here - http://www.checkthishouse.com/6290/room-humidity-and-house-humidifier-energy-savings.html
Check your foundation for any possible seepage, also, just to make sure that your carpeting and padding are not contaminated, lift it up in several spots along the foundation. If you notice any black residue on it (especially on padding), well, you may need to replace it.
Since you have a sump pump, maybe even ejector pump in basement area – seal the covers air tight.
If you have older cast iron or galvanized drain pipes – they often corrode and crack, sometimes along the top on horizontal runs. In such cases you may not see it but it will release moisture and unpleasant odors - http://www.checkthishouse.com/1757/cast-iron-drain-lines-problems.html
You may also have a small plumbing leak above one of the air ducts. This may not be visible from underneath but it could result in corrosion and water leaking into the duct.
Those are all guesses but maybe one of them will be the right one J.