There is a good chance that you are seeing groundwater entering your basement from cracks in the floor. The level of groundwater is determined by a combination of local topography (the grade of land around your house), the precipitation, time of year, and soil properties. The best way to handle groundwater is to install a perimeter drain around the house, and direct the flow away from your basement, typically to a low spot on the lot.
It is possible that your house does not have a perimeter drain, many houses do not (I don't have one at my house). Unfortunately, it was not common to install perimeter drains 30 or more years ago, only recently has it become common practice.
The sump pumps will only reduce groundwater to the level the pumps draw down to, so for example if the low level cutoff is 3 inches below your slab, groundwater cannot be lowered more than 3 inches below your slab using the pumps. Additionally, the pumps only draw from a limited area. The lower the permeability of the soil beneath your slab, the smaller the area the pumps will drain.
There are some things you can do yourself, and others that may require professional assistance. Installation of a perimeter drain is a relatively simple project, however you need a backhoe and a good design for the drain. You also need a suitable place to direct the water to. If you can get a backhoe, or hire an operator, you could do some or most of this work yourself. A perimeter drain is most likely to eliminate your problem. If you wish to pursue this option, I could give you some ideas about how to construct one.
You could lower the sump pump pits. This would allow the pumps to draw the groundwater down further, and would potentially eliminate your inflow problem The further you lower the pump pit, the more often the pumps will run. Lowering the pits typically requires you to dig a deeper hole, which is harder than it sounds when the hole is full of water. This is a project best undertaken during the dry season.
There are also interior drain systems that are typically installed by professionals. They are typically pretty expensive, but can be very effective.
Attempting to waterproof your floor from the inside is almost impossible, and even if it worked, the pressure of the water underneath the floor could crack the concrete. I don't recommend it.