The thermohaline current in the North Atlantic ocean sinks because it gets cooled, or it gets cooled because of sinking?
Complementary to the points about salinity emphasised by TomO and S Verhoef, one more point to emphasise is the role of deep ocean convection. You may have in mind this picture of water sinking because it gets heavy, which is not quite wrong, but you get a more accurate picture if you see that the water column down to about 3000 meter deep is convectively mixing (mainly in winter) in localised areas of the Greenland, Iceland, Norwegian and Labrador seas. Convection is what you get when dense water lies above lighter fluid, which is as unstable as an egg on its head.
The consequence of this mixing is twofold.
First it shapes the hydrological structure of the deep ocean and determines its organisation into distinct water masses (e.g.: http://www.soes.soton.ac.uk/teaching/courses/oa631/plot_20w.jpg )
Second it induces a strong cooling of the ocean water masses.
Together these effects control the strength of the meridional overturning cell.
If your are after an advanced and authoritative review on this, see http://onlinelibrary.wiley.com/doi/10.1002/2015RG000493/full, which is open access.
It not only is cooled but saltwater sea ice separates fresh water in the ice forming heavier brine that drives the cold water circulation.
The thermohaline circulation is what you mean I suppose.
The water cools down while traveling North, causing it to become more dense. Like TomO tells, the water gets saltier due to the loss of salt from the ice mass*. The density of the water will therefore increase, resulting in the sinking of water parcels.
*Salt will be released from the sea-ice due to the difference in freezing point of salty water and fresh water. If the temp rises to just below 0*C fresh water will still be frozen, but (dependent on the salinity) the salty ice will melt due to the lower melting point of salty water. Therefore sea-ice will become fresh after a while, and brine is released into the seawater making it more saline.