There are two ways of approaching this question — using biblical chronological information to establish an approximate date for the event, or using whatever archaeological information is available that might suggest a destruction date.
Unfortunately there is very little archaeological evidence available, although that hasn’t stopped people using what is there to suggest possible dates. Even the location of Sodom and Gomorrah is disputed. Possible sites include Bab edh-Dhra, Numeira, es-Safi, Feifeh, Khanazir and Tall el-Hammam. The evidence across the various sites is scanty and there are several possible layers indicating burning and traces of sulphur. For the most likely possible sites, the date of destruction is around 2300BC. Bryant Wood (1999) provides a good summary of the available archaeological evidence.
The biblical chronological evidence is substantially more detailed, and enables a date of about 1845 BC to be determined. This is based on an Exodus in the mid 15th century BC, and then working backwards to obtain a date of about 1946BC for the birth of Abraham (see Hyndman, 2006, chapter 3). We know that the destruction of Sodom was one year before Isaac was born (Genesis 18:10,16-33). As Abraham was 100 years old when Isaac was born (Genesis 21:5), that gives a date of 1845 for the destruction of Sodom.
It is not clear how to reconcile these figures. Perhaps the cities being excavated are not Sodom and Gomorrah, or maybe the archaeological dating is in error, or possibly the biblical data has been misinterpreted in some way.