Can geckos climb a wet surface?
Short answer
Yes they can, but with less efficiency. 

Background
Efficiency of sticking to a glass surface decreases about 20-fold when the gecko's feet are wet (Stark et al., 2012). On dry glass, Gekko gecko holds on with about 18 N, which is about 18 times their own body weight. Hence, the gecko attachment system is over-designed (Knight, 2012). Wetted toe pads reduce the adhesive force significantly down to 0.9 N. A wetted glass surface does not significantly affect the grip of Gekko gecko, except after the gecko takes multiple steps. Hence, geckos can cling to wet surfaces, but only when their feet are not getting too wet. 

Note that gecko attachment is not based on suction, such as the suction cups seen in octopus. Instead, geckos cling to surfaces with Van der Waals forces. Van-der-Waals-based adhesive systems take advantage of weak intermolecular forces between two surfaces. Upside is that these forces are not surface specific, allowing geckos to climb trees, rocks, walls and even glass windows. 

Geckos achieve their strong attractive force to the substrate by utilizing small hair-like structures on their toes called setae, which are made primarily of keratin (Fig. 1). Setae contain flexible spatulae which make intimate contact with rough and even dirty surfaces. 

  
Fig. 1. Left panel: Close-up of the underside of a gecko's foot as it walks on vertical glass. Source: Wikipedia. Right panel: Images of (A) dry and (B) wet setae using the same digit of an individual tokay gecko (Gekko gecko). The image in A was taken prior to exposure to water and the image in B was taken after 30 min of soaking in water. Source: (Stark et al., 2012).

The drawback of using Van-der-Waals forces for grip is that the Van-der-Waals forces only exist across surfaces very close together, and an intimate contact between the spatulae and the substrate is required for van der Waals forces to be effective in the gecko adhesive system (Stark et al., 2012). Wetting of the feet decreases the Van der Waals interaction with the surface.

References
- Knight, J Exp Biol 215, i
- Stark et al., J Exp Biol; 215: 3080-6