The Huang Gang (皇崗) (name of the border at Shenzhen side) / Lok Ma Chau (落馬洲) (name of the border at Hong Kong side) border crossing is the only 24-hr border crossing between Shenzhen and Hong Kong. So if you will arrive late at either of the cities, you will have to use this border crossing.
If you land in Shenzhen airport late at night and want to go to Hong Kong, it is most convenient to take a taxi to the border. It will cost about RMB130 by taxi and take about 45 minutes.
Once you have gone through the Shenzhen control point, you will need to take a shuttle bus (called yellow bus) at HK$7 to the Hong Kong Lok Ma Chau Control point. Then you will land at a traffic interchange. Here you can find 24-hr transportation to the city, such as the 616S night bus going to Mongkok every 20 minutes.
Also, every day, there are 24-hr buses going between the border and Wanchai Pier/MTR Prince Station/MTR Tsuen Wan Station. The bus fare is about HK$40 per journey.
If you plan to take a taxi from the border to Kowloon or Hong Kong Island, do remember to take red taxis. Only red taxis go to these two regions. The green taxis go only around New Territories, where Lok Ma Chau belongs. It will cost about HK$300 by taxi to Tsim Sha Tsui, or HK$350-400 to Central.