I have detected noise at the driver input, so maybe the problem is not the path from driver to mosfets, but the path from pulse generator to driver. How can I clean it? Using a low pass filter?
That looks pretty much like a ground layout problem ...
Yes, you'd need to improve your ground connections from the pulse source to the driver to the MOSFET itself. A gate resistor of appropriate value (usually in the 1R to 10R range) often helps a lot. If haven't done that yet, make your ground a plane including MOSFET source, driver GND pin and pulse source GND pin. Keep the driver as close as possible to the MOSFET (think in millimetres here). If the driver picks up noise at its input and you cannot improve your layout, consider opto-isolating the pulse source from the driver, and place these three parts (optocoupler, driver and MOSFET) as close as possible together.
Often such oscillation results from ground bouncing (the drivers GND pin gets "lifted" against the pulse source output GND). Proper grounding helps here, so I'm repeating: Put everything as close as possible and make use of a ground plane.
A gate resistor that is too high won't work too,
Edit: you mentioned "short cables" somewhere, is this a breadboard like prototype?
If so, forget about it. It just wont work. This kind of circuit requires a proper layout or at least a veroboard (perf board) construction with real short (millimetres) connections.
See this example of a perfboard circuit I've made some time ago, this is a MOSFET pulser creating 800V / 5ns fall time pulses - works quite well. Located bottom left is an optocoupler and next to it the SO8 MOSFET driver. The driven mosfet is located directly above the driver chip (but on the other side of the board). Note the solder wick that I used as GND in this area, and some parallel wires to lower the ground impedance to the SMA connector.
See also here, this is a very unfinished (indeed not yet started) page about it:
http://wunderkis.de/pulser/