You want to use the "Formula" entry for the workflow entry criteria.
When you are creating the workflow, change the rule entry to Formula Evaluates to true - and then use the formula function ISPICKVAL.
In your case, enter the formula
This means your workflow will only be executed on records when this pikclist value equals your string literal.
ISPICKVAL is defined as:
ISPICKVAL(picklist_field, text_literal) Checks whether the value of a
picklist field is equal to a string literal
And further information on it can be found in the documentation here.
Once you have set up your entry criteria for this workflow, you will need to make a new "Field Update" action for the process. Click this in the immediate actions area.
What you want to do is update your date field, so select it from the drop down, and then, in a second, the "Value" section appears, switch to formula view again and enter "NOW()" like so:
Save your action, save your rule activate the workflow!!! and you should be roughly good to go?