What is the deepest accessible point on earth?

If you're looking for deepest place on the surface it would be the Dead Sea which is -427m deep.

If you're looking for the Ocean one there is a tourism organized by Deep Ocean Expeditions, which you can pay a pretty penny but explore like James Cameron down to the depth of the Mariana Trench...

If you're looking for something more or less reasonable as far as prices are concerned take a look at R.I.D.E.

What you are probably looking for is this list https://en.wikipedia.org/wiki/Extreme_points_of_Earth#Elevation

Which tells us that, apart from the dead sea,

• lowest human-sized point underground is 3.9 kilometres (2.4 mi)[1] below ground at the TauTona Mine, Carletonville, South Africa

Google shows the entrance at about 1700m which would make the bottom 2.2km below ground you would have to go to sea to beat that as Karlson suggested.

If you also mean the deepest point underground: The deepest natural cave is the Voronya/Krubera Cave in Abchasia near Georgia/Russia with an estimated depth of over 2000 meter (6500 ft). The entrance is in 2200 m height so you are still over water.

I have used this as background in a RPG game and searched photos for it. It contains very long shafts, an underground river...it is quite a fantastic cave.

Deepest human accessible point in the world is the bottom of Kidd Mine in Timmins, Canada. Kidd is 9,889 ft (3.014 km) deep and the city itself is only 100m above sea level. http://en.wikipedia.org/wiki/Kidd_Mine

I don't think it's the deepest, but the LKAB Visitor Centre in Kiruna/Giron, Sweden, is easily accessible by daily buses, and is, if I recall correctly, 345 metre below the local surface. Not nearly as deep as caves posted in other answers, but a lot more accessible for the average tourist. The mining is currently 1385 metre below the surface, but is not open to regular visitors.