I'm not sure you can do it with one object. You will need the text box itself and a masking object.
I'd put any border around the actual text box, then put the masking object, filled with white, behind it and adjust the opacity to get the effect you want. That's what's commonly used in magazines, brochures, etc. You can use some of the Effects to get a slightly different result, but I am pretty sure you can't make the box one thing (translucent or blurred) while the text remains sharp, not in one object.
You could also place and blur the text background in the image itself, in Photoshop, but that's harder to tweak.