In general, roofing companies are not experts in diagnosing "water in basement" problems; however, I am an experienced house builder and remodeler, so I will provide you with my knowledge on this subject.
If the source of the water problem is not an obvious water-flow issue, for example, water flowing down a sloped driveway and directly into the garage, then the problem is quite often the result of excessive ground water next to the foundation.
Because of gravity, the water levels on BOTH sides of the concrete (inside and outside the house) want to equalize. If the water level on the outside of the house is higher than your basement floor, then you can have leaks because the water level inside your house will want to rise until equilibrium is achieved.
Modern building construction makes great efforts, therefore, to remove excessive water from the outside of the house. This is done in two ways. First of all, modern houses have a footing drain. This is installed at the height of the footing, which should be about 4-6" lower than a finished concrete basement floor. This drain collects water from the perimeter of the house, then takes it elsewhere (typically to a combination sewer, a storm sewer, or "daylights" to a city street).
The second type of drain on a modern house is used to collect all roof water, and direct this water away from the house (again, to a combination sewer, a storm sewer, or "daylight").
The first thing you should check is your downspouts. Does your roof water drain next to the house?? If so, this needs to be changed as soon as possible. Install drain lines that will collect the roof water, then direct this water away from your house. IMPORTANT: Do not direct this water towards a neighbor's house, as you might flood them out, nor should you direct the water to the top of a steep hillside, as the hillside may collapse due to excessive water!
If this does not solve the problem, you might have to install a sump pump in your basement. Typically, these are dug down 18" to 24" lower than your basement floor. By having it recessed, you are encouraging ground water to flow to this lower point. Inside the sump hole, install a pump, and pump the water out (and away) from your house. A sump pump is a good alternative for older houses since it would be virtually impossible to retrofit a house with a footing drain. Call a plumber for more information on sump pumps and sump pump installation.