This is a well known issue of the game since its release.
See here: http://www.saintsrow.com/community/go/thread/view/136781/28692595/Lag_while_in_co-op_lan
The problem is not in your hardware, system or network.
The problem is in the game engine. But neither the game developer - Volition, nor THQ -publisher, even commented on the issue yet, let alone taking any effort to fix it.
(There is a single, one line response from sombody from Volition on the mentioned forum, and that blames the issue, quite nonsensicaly, on somebodys low hardware specs (!?) ).
There are apparently hundreds of people with varying HW and network connections, including me. Including people with flawless lan\net paramaters, people who exceed HW specs for this game and its co-op. Still having the issue.
This is a state of things well in 2013. Still no real fix (official or unofficial) is known to me :(
There are some hints on the forum on how to make it more bearable. In short:
Try to make the game run with same or as similar as possible frame rate (FPS) on all computers that are connected to one coop session.
That can be partially (but tediously) achieved by tweaking ingame - mainly Graphic settings.
E.G. On faster computer set graphic details higher (even ridiculously / needlessly high) to slow the game down. On slow computer lower game resolution and other graphic effects settings. Do so until the game on both computers run with as similar FPS as possible.
IF your net connection is all right, this should reduce (though not remove) the rubber-banding effect, at least while driving cars.
This is what I found it the mentioned forum. I didn't test it myself (because the process is really tiresome and not worth it for me).
The bottom line is: remember this issue and how it was never fixed, before buying next game from either of the above mentioned companies.