That's a single bowl sink, correct? If you are getting a garbage disposal, you don't need a strainer basket. Otherwise, I think I would match the faucet.

Hee hee heee, rotflmao!!!!! Forgive me,you said no laughing.
Satin nickle is throughout the metal and will never peel like chrome.

I am not laughing because I asked the same question when I was doing my reno.

Chrome all the way. (Peeling? Come on. Chrome doesn't peel.) You won't notice any difference when it's at the bottom of your sink. I have a brushed stainless sink and a polished chrome strainer (this includes the flange around it) and it looks absolutely fine together.

I'm pretty sure my drain came with a strainer basket...

If you can, get a SS strainer basket/drain (our sink, Julien, came with its own SS drain and strainer basket). If you get satin nickel, and clean your sink with BKF for example, the satin nickel can tarnish. SS won't.

I would just keep it simple and get stainless steel drain flange and stainless steel basket. Match the sink. One less problem should you ever replace the faucet in a different finish later on.

Do not get a satin nickel strainer (or anything stain nickel inside the sink). Satin nickel will corrode if you use pretty much anything other than hand soap or hand dishwashing detergent to clean the sink or clean items in the sink. Bleach, Bar Keepers Friend (and similar), etc. will all damage satin nickel. How do I know this? *sigh* experience.

I have brushed stainless steel in my kitchen sinks (w/Satin Nickel faucets, soap dispensers, and air switch. However, I forgot to specify the same in our Powder Room and our KD put in not just a satin nickel faucet, but also a satin nickel drain - and it's now damaged from using BKF from cleaning.

My recommendation is neither chrome nor satin nickel - instead, get stainless steel - brushed stainless steel. (if the chrome is brushed, it's probably fine.)

As to using the strainer that comes with some sinks (not all include the strainer), we opted to not use it and, instead, got the Kohler DuoStrainer. The "free" ones are usually pretty cheap - the DuoStrainer is heavy-duty and locks both open and closed so it never closes on you when you want it open (or vice versa).

Nickel is the metal added to steel to create stainless steel. If any nickel fixture corrodes, it is because it has been plated to an inferior metal. The point is, don't buy cheap fixtures.

We don't have a garbage disposal so there is almost always a strainer in the bottom of our sink. I just buy inexpensive ones and once or twice a year replace the strainer.

Your mileage may vary.

Garbage disposals are just plain wrong.

Thank you all. This is been very helpful. We don't have a garbage disposal we have a cesspool so my husband doesn't like the idea I'm used to not having one anyways.
I ended up ordering the satin nickel strainer basket that is made by Franke.
I'm a little nervous given prior poster's experiences
But time will tell - I do like using BKF on my pots
And current sink....so hopefully I will not regret
The decision.
Thanks all for the input.