NASA is an agency of the American government. It could cease to exist at any time (at least, on a long-time scale). Would they even still be around by the time humans get to Mars? There are, on Earth, two countries that can still send humans in space (albeit not very far away) and a half-dozen with space agencies that COULD, if they were pressed, eventually send humans in space. A collaboration of countries could be able to send humans to Mars within two decades.
And there are some private enterprises (yes, word chosen on purpose) that are working on sending humans in space.
Very few of the above need NASA.
If, however, you want to know if humans even go beyond Mars (regardless of the agencies or countries that sends them there), now that is a different question.
There might be two driving forces:
1) exploration (humans have always been explorers): once "we" reach Mars, someone will have the dream of going further. If the effort is not interrupted by one of our endless wars - or by humans erasing themselves from the face of this planet - then it is inevitable that "some day", humans will go somewhere beyond Mars.
2) greed: if someone finds a way to make tons of money by going "out there" (to mine asteroids? to sell land on Ganymede?), then they will also find a way to get there. With or without NASA.