What's "watering place" in this case?

In this context, watering place almost certainly means a drinking establishment, such as a public house or hotel bar.

The Adventure of the Dancing Men is set in Norfolk, where the coastline is low and marshy, very unlike the southern and western coasts where seaside resorts are more commonly located. It’s not that they don’t exist, but the link below shows the Top 10 in contemporary Norfolk, and most of these are not greatly developed.

https://www.visitnorfolk.co.uk/inspire/Top-10-beaches.aspx

However, the real clue lies in Hilton Cubitt being married to Elsie Patrick, an American woman whose past included membership in a criminal gang, from which the “dancing men cypher” originally came. The use of “watering place” as a euphemism for a bar is common in the American writing of that time. The most likely explanation is that Conan Doyle was intentionally writing in an American idiom, even though the specific words in the story were spoken by an Englishman.

Edited to add:

In the story, which is available on Project Gutenberg, Hilton Cubitt is described as the heir to Riding Thorpe Manor. In order to get there, Holmes and Watson take the train to North Walsham, and then travel by horse and carriage to a point from which the German Ocean (now better known in English as the North Sea) can be just seen. There is a town in that general area called Edingthorpe. Thorpe is a variant of the Middle English word thorp, meaning hamlet or small village.

The nearby town of Mundesley-on-Sea was a popular seaside resort in Victorian days. However, it also had three notable drinking establishments, the Ship Inn, the Manor Hotel, and the Royal Hotel. Sir Arthur might have had both these meanings in mind, and used “watering places” as a sly euphemism for his readers to enjoy.