I'd start at the sump pump basin. Sounds like it isn't being used. The joints for the lines going into the basin could be leaking and need resealing. I assume this is not a sewage pump basin or else that could be the problem. Do you have a basement drain somewhere and where does the main sewer line exit the basement as far as being above or below basement floor level? All basement drains should have a trap(to hold the sewage odor in) and oftentimes the water in the trap will evaporate from lack of use and if that is the case then you either need to run water down the drain regularly or fill the trap with something like vegetable oil that won't evaporate.
If it is a wet ground/moisture problem under the slab and maybe they didn't put down a plastic vapor barrier, then try a dehumidifier down there.
Sometimes in the spring and fall you get some air temperature inversions or the wind that cause a downdraft down plumbing vents and the flues for the furnace and gas water heaters which could account for the gassy smell. Fireplace chimneys are known to downdraft also.
Sometimes hot water can get a rotten egg smell in it.