I got a super deal on some 17 by 8 inch wheel. Was wondering if 225 45 17 inch tire will fit on my 2003 Mini Cooper base. The offset on the wheels is a 38.

 

might need spacers u will be 3mm closer to your sturt i got 17x8x5 20et with a 205/45/17 on mine

 

Short answer: No

Compared to OEM 16x6.5 et 48

17x8" et38 will be 9mm closer to inner suspension and poke out 29.1mm more.

225/45-17 is too large for a tire since it is both wider and taller than OEM and you want to put it on a wide wheel with aggressive offset.

It could work if you used wide fender arches or cut away the black plastic wheel arches but that's not going to be practical.

Standard tire for 17" is=
205/45-17 or 215/45-17
These can be stretch fit onto a wide 8" rim but it won't look like a correct fit and tire wear will be faster since fitment is not ideal.

Note the OP has a base cooper from 2003 R50. The clearance for wider and taller tires/wheels is limited more than recent models.

 

I run 235/40/17 on 7.5 rim with 45 offset.
The key to all fitment issues is offset and backspacing.
I'll venture you should try a 5mm spacer to put your offset @ 43 and go from there.
Remember to measure thrice; buy once.
ISAMIN