Yes, airborne infantry took prisoners. Looking at two of the major airborne operations of the war:

During Operation Merkur, the German invasion of Crete, German paratroops captured Allied troops from the second day of the battle, May 21st. Some prisoners were put to work at Maleme airfield (in contravention of the Geneva Convention), others were taken to holding area on the western bank of the Tavronitis River. (Details from Prisoners of War, W. Wynne Mason, part of The Official History of New Zealand in the Second World War 1939–1945.) With the German capture of the island, prisoners were held in camps on the island, then later moved to Austria and Germany via Greece.

During Operation Market Garden, British airborne infantry around Arnhem took many German prisoners (as seen in e.g. [an Imperial War Museum photograph] (http://www.iwm.org.uk/collections/item/object/205192050)). 2-300 were held in a compound on the tennis courts of the Hartenstein Hotel, Oosterbeek (the hotel itself was used as the headquarters for 1st Airborne Division); it can be seen in [a short film clip] (http://film.iwmcollections.org.uk/record/index/21592) from the IWM. When the 1st Airborne were forced to withdraw the prisoners remained at the hotel, along with wounded British troops, to be released/captured by the Germans.