So, first of all, what does it take to send the signal in the first place? Each of the Deep Space Network sites has a 34m antenna that is typically used for inner solar system communications. The power is approximately 200- 400W, in a band where that much power is difficult to achieve. In order to jam the signal, you need to increase the noise significantly. Overall, it would be rather difficult to jam the signal going to the spacecraft. It could be done, but building dishes of the required size is a tricky matter, although I don't doubt that a dedicated source could make this happen, given the appropriate desire.
Okay, so what about the return signal? Well, that one is a bit easier, as the signal is rather weak on the ground. I don't have the full rf budget of a mission, but it is safe to assume it is weak. However, this is going to have some serious difficulties. First of all, the antenna is very directional, very little RF will get in to it except in the direct path of the antenna beam. In fact, from this paper, the feed horn, which is essentially the receiver, at 22 degrees off of the beam there is a 38 dB, and that is the best sidelobe! If the horn is properly shielded, as is likely being in the dish, then the side lobs drop down dramatically. Even if it is not properly shielded, while it would be possible to jam this signal if you can find the correct location to do it from, it would have to get pretty close to the direction the antenna is pointed to take advantage of that. In addition, the bands of interest are essentially line of site only. You would have to get close to the antenna to have any success, and the signal would lead directly to the device that was transmitting. It could theoretically be done, but it would be difficult to achieve.
Bottom line is, the most likely scenario for jamming would leave the guilty party quickly found. All spacecraft I'm aware of have a method to re-transmit lost data. It would be painful for NASA, but of limited damage, and the person would end up in jail rather quickly, no doubt.