What does "steenking" mean?

It's supposed to be a Mexican accent.

From the wikipedia article you linked:

In one scene, a Mexican bandit leader named "Gold Hat" (portrayed by Alfonso Bedoya) tries to convince Fred C. Dobbs (Humphrey Bogart) that he and his company are Federales:

Dobbs: "If you're the police, then where are your badges?"

Gold Hat: "Badges? We ain't got no badges. We don't need no badges. I don't have to show you any stinkin' badges!"

Both a 1967 episode of The Monkees and Mel Brooks' 1974 Western Blazing Saddles misquoted the line as "Badges? We don't need no stinking badges.", and it is this version of the phrase that went on to be quoted, paraphrased, and parodied in a variety of subsequent media.

It's more obvious if you watch the clips in question.

Here is the original clip on YouTube:

And here is the Blazing Saddles version:

The "steenking" you found is just the phonetic spelling of the accent used by these actors.

Note: Whether this accent is accurate or politically correct or insensitive or funny I leave up to you to decide. But for better or worse, this is what whoever wrote that comment was referencing.