In this article is stated that in order to land on Mars NASA might need the help of the ESA or even the Chinese, not because it doesn't have the capability, but because it needs more financial help. Of course for the moment this collaboration might get little traction in the US government because of security reasons. Although little publicized, it appears that NASA accepts donations, but you can't suggest/know what they do with your money. Probably this type of approach doesn't create a sense of accomplishment within the people that manage to find out they can actually donate to NASA. My question is what legal biases prevents NASA from launching targeted, well advertised campaigns (kickstarter like), that could raise a part of the money needed for specific missions?
NASA Policy Directive
COMPLIANCE IS MANDATORY
Effective Date: April 28, 2010
Subject: Acceptance and Use of Monetary Gifts and Donations
Responsible Office: Office of the Chief Financial Officer
- POLICY The National Aeronautics and Space Administration (NASA) may accept and utilize monetary gifts, donations, or bequests given as cash, check, or money order, provided they are unsolicited and offered without conditions on their use.
To expand on the currently accepted answer: The "unconditional gifts only" rule is not NASA-driven policy, it's a matter of federal law. Specifically, three laws control this. The Miscellaneous Receipts Act requires that any money that a federal agency receives, absent any other statutory authority, has to go into the general treasury -- unless a federal law specifically allows an agency to collect money, whatever you give them is treated as a gift to the United States. The Antideficiency Act says that an agency can't spend money that was not appropriated for it under most circumstances (important exception: agency trust funds). So, for a generic federal agency, if you donate money they have to forward it to the general treasury. If you donate money to the US under the condition it go to that agency, that condition is binding, but just means the government can't accept the gift without Congressional authorization (as Congress must approve all gifts that impose some duty on the US).
Now, NASA is not a generic federal agency: they have specific statutory authority to accept gifts. But that authority only goes so far as unconditional gifts -- they can be accepted and are auto-appropriated for NASA by going into the NASA trust fund, but nothing else can be accepted. They can also enter into contracts, but they can't make profit off those contracts - it's limited to reimbursement. So, because NASA lacks statutory authority to accept conditional gifts, they cannot do so without the consent of Congress -- otherwise it'd involve putting money in their trust fund that they have no authority to keep, as well as spending money they have not been appropriated.
Incidentally, for why federal agencies need special authority to accept gifts, there are various solid public policy reasons involving checks and balances. But that's not on-topic here: the point is it's illegal for NASA to accept conditional gifts, not because of their own policy, but by act of Congress.
As noted, NASA cannot campaign for donations. However, even if someone else does the campaigning for them (e.g. the Planetary Society), the amount of funds you might expect to get are small in comparison to what is needed for a mission. The Planetary Society managed to raise \$4M at one point to build a solar sail spacecraft, which is quite impressive, but down about three orders of magnitude from NASA annual spending on space science. Similarly, the largest kickstarter campaign so far raised \$13M (for an advanced beverage cooler).
Another common question along these lines is selling advertising on spacecraft. The Coca Cola or Nike logo on Mars might be worth something to them. It turns out that the amounts are quite similar, where such an opportunity would be worth a few \$M to at the very most a small integer times \$10M. Note that a Super Bowl 60-second spot costs \$8M, and that will likely have many more viewers than a Mars mission.
By the way, since you're keen on this, the Planetary Society's Light Sail 1 is scheduled to launch in 2016, and they are accepting donations here. NASA just cancelled their project to launch a solar sail.