According to Rich Cimini:
More precisely, the New York Jets have $23,815,532.00 in space, according to the NFL Players Association, but Mr. Cimini's number is close enough. However, that number doesn't truly reflect how much the team has, because a portion of that number is already allocated elsewhere.
According to Jason Fitzgerald at OverTheCap, the incoming rookie draft class, assuming the team uses all twelve picks, will cost $7,353,384 in salary cap space (see here for more detailed information on how that's calculated on a per player basis). That number, however, will be reduced to $2,313,384 in actual cap space once you remove the players currently on the roster. This has caused some confusion, so let me explain:
The roster is currently composed exclusively of veterans (obviously, since the draft hasn't taken place yet). They are making a minimum salary that can be substantially higher than rookies (anyone drafted after the fourth round makes less than the lowest veteran's minimum). The value of their minimum depends on how many years they have been in the league.
An estimated six drafted rookies in this year's draft class will make less than the lowest paid veteran, and that's not even including undrafted free agents. So you have the original number of $7.3M, but that gets reduced to a practical number of $2.3M when you account for the rookies that offset the displaced veterans when you try to make the 53 man roster. If both players are making the minimum, that's can be up to a $535,000 gain by swapping a veteran for a rookie. Swap a few players, and we're talking serious money, which explains why the $7.3M hit falls to just the projected $2.3M.
That brings the team down to $21,502,148 in free space. But teams allocate a certain portion of their remaining space to in-season transactions. If a player gets hurt and they need a replacement, that comes from the allotment. This is obviously a loose figure and can change dramatically by the team, but a fair estimate is $2-3M. For simplicity, let's call it $2.5M, and we'll assume the team uses all of it, even if they won't.
That leaves, roughly, $19,002,148 in free space heading into the 2014 NFL Draft. As undrafted free agents and regular free agents are signed, and other players are released, that number will continue to fluctuate. However, whatever remains will be rolled over into next year's salary cap space, which leaves the team flush with cash for the foreseeable future.