Borders provide economic stability.
In any nation, there is a limited number of resources: vehicles on the road that the city infrastructure can support, public school capacities, and hospital beds. Without borders, mass immigration would overflow the nation and its infrastructure, leading to total disorder.
The European Union's open borders and lessened immigration restrictions have economic downsides. Locals cannot compete with cheap foreign labor. Places to live are harder and more expensive to come across. Eastern European countries are facing a shortage of doctors and nurses because they opt to move west for better economic opportunities.
While open borders may benefit skilled migrants who are free to move where there are better opportunities, countries with these better opportunities cannot sustain the influx of immigrants.
Borders are needed to provide geographic boundaries of national sovereignty. Borders help define who can receive state benefits, and who is under the government's jurisdiction. The people's social contract with each other and with the government cannot exist without borders, immigration restrictions, and enforced rules for attaining citizenship. 
Without borders, governments could potentially claim jurisdiction to intervene in any country's space, which would cause more chaos than border disputes do today.
Nation-states must have borders to function properly. Democracy flourishes when its citizens can play a role in the political process, and maintaining borders helps define who can or cannot participate in a country's political functions.