Full stop here: We all do it, even when we don't think we are. It's a matter of fact however that doesn't make it right nor does it mean we should continue.
First, let's take a look at what stereotyping is.
A stereotype is defined as an often unfair and untrue belief that many people have about all people or things with a particular characteristic (Miriam Webster Dictionary). This can be how we see someone of another gender (ie: how men see women or women see men... or even how cisgenders see transgenders and vice versa), what assumptions we make about people of other races, economic situation, where they are from in the world, or where or how they were raised, what faith they are or are not and then assuming how that affects their personalities and choices in life simply based on it.
I'm pretty sure you can see how this could get you into trouble outside of writing - say, at your day to day job or even just talking to someone on the street, or if this affected how you treated someone else...
Side note: Character archetypes are NOT the same as stereotypes. Those are roles (think of them as job descriptions) within a story where the individual character traits are as fluid as they are in real life. However, it doesn't mean that you can't fall into the same trap of creating the stereotypical "hero" when writing... it's really easy to do this. Too easy, but I digress.
So, why shouldn't you do it?
In life, like in art, there are numerous reasons why you shouldn't stereotype people, or let your characters be a stereotype. The primary reason is... people don't like other people to assume things about them. Think about how you would feel if I judged you before I have even met you based on how you look or what you took in school, or even what school you went to? Or, by what car you drive or where you live. Sometimes, it could end up to your favour where I like you because you drive a sensible, but perky, car like a sportier Honda. But what if, if you didn't own or drive that Honda I automatically decided I didn't like you because I assumed that we wouldn't have anything in common simply based on it?
How would you feel?
I'm willing to bet you wouldn't like that very much. Maybe you wouldn't care if I liked you or not, but what if it was someone you were trying to get to know... or what if you got turned down for that job you really needed to have so you could pay rent because of it?
How do you feel about it now?
I can already see what some of the writers are saying. But, Kristan, I get that. How does that affect what I'm writing?
Now, imagine one more step here where you allowed your views on people--your prejudices and the stereotypes--to bleed through when you create a character. You create a character based on something you assumed because of a racial stereotype... or even a gender stereotype. What if the character you created is the "blonde bimbo" because she's a) blonde and b) happens to be Californian. This was a very common stereotype (that many people bought into) in the 80's that still pervades conversations today. I have a family member (no names) that calls another family member dumb, but also adds the words "dumb blonde" and even says she's dumb because she's blonde.
All of this because of a series of characters created in the 80's about blondes not being the most intelligent bean in the bowl.
Which, I might add, is completely untrue but the stereotype still remains.
That, ladies and gentlemen, is why we do not want to fall into the trap of allowing, or furthering, stereotypes in our writing.
STEREOTYPES, ONCE ESTABLISHED, STICK AROUND FOR DECADES.
In a writing group, my post in relation to stereotypes was actually deleted because someone thought it would lead to people being hateful. I can understand that because this subject is by far not the most comfortable to discuss. People don't want to let go of stereotypes for many reasons but this is an issue (like racism, cultural appropriation, and gender issues) that needs to be tackled head on because it leads to those other issues by not being seen as an issue because the stereotype is familiar, like an old friend or blanket, and makes it easier to see the other issues as a non-issue, especially if you are not the one being affected by the stereotype directly.
We need to tackle stereotypes head on, even in fiction, and call them out as unacceptable.
Enough of the super buff jock to sweep in to save the damsel in distress, but needs the same damsel to sort out his finances because, like the "dumb blonde", he's a jock and is therefore dumb. Enough of the nerdy fellow in computer sciences because only the skinny guy with oversized glasses is intelligent but can't seem to figure out his feet. Enough of the dumb blonde. Seriously... Enough.
We can do better. We have to do better than this. It seems harmless, but if you leaf through the numerous studies on the subject (or refer back to Exhibit A: the "Dumb Blonde/Dumb Jock") you will find that allowing ourselves to fall back to them we are doing far more harm than good.
References:Penn State Civic Studies Blog
United Nations Human Rights: Office of the High Commissioner Blog
Dr. Nathaniel Lambert Ph.D
The Guardian Today
Quora - Another discussion on stereotyping
Debate on Debate.com
Bright Hub Education
April Kemick, University of Toronto
Article for USNews
The Stereotype Threat
Facing History and Ourselves