I need to find potential routes for undersea power cables in South East Asia, therefore I need to select all areas where the ocean is less than 4000m deep. I load the "ocean basemap" from Esri but it looks like it is only a picture and no attribute table with data about how deep the ocean is at an specific point. Is there a way to create a attribute table out of a map or do I have the wrong map?
As Stefan says, you have the wrong data, the basemap is just an image with no elevation data.
A quick an easy way would be to download the 0m and 4000m Bathymetry data sets from Natural Earth, and clip the 0m shapefile with the 4000m shapefile (use Erase tool in ArcGIS or Difference in QGIS).
Or if you need a more detailed raster data set I'd recommend SRTM 30m data from the USGS.
You have the wrong data. The ESRI Ocean Basemap is designed as basemap. It's also cached, so you only have the image tiles available and therefore it cannot be used for analysis.
You need to get a spatial dataset with x, y & z values in order to perform depth analysis.