What properties does a solvent with a high dielectric constant have?
Before I start, please note that I am not a chemist so be sure to verify what I say...

The electric conductivity of water depends on what is dissolved in it. For example, salt water will be far more conductive than distilled water. So if you're professor was talking about sea water, the answer could actually be different from purified water.

Due to the dissolved salt ions, sea water is a better conductor of electricity and so by definition it will be a poor dielectric, since according to this Wikipedia article: to be a good dielectric a substance must be both a good insulator and able to be polarized by an electric field.

Purified water on the other hand will be a better dielectric because it does not contain any significant amount of ions and is therefor not very conductive (i.e. is a better insulator than sea water).

I think (but don't quote me on an exam) that the reason alcohol is a poor dielectric is because it is not nearly as polar as water is, and according to the above, the second requirement for a dielectric is that it could be polarized by an electric field.

(This might also have to do with the size of the molecules, since water molecules are generally smaller than many kinds of alcohols, and can therefore re-orient themselves more easily in the presence of an electric field, but this is a guess.)

The Solvent Classifications section of this Wikipedia page also mentions that the polarity of a solution can be roughly gauged by the solution's dielectric constant. So this too seems to support the idea that polarity and dielectric properties are related.

If anyone has any improvements or corrections, please let me know in the comments and I'll update the answer accordingly.