I vote "Teach" despite the fact it is not always wrong to split an infinitive.
Infinitives are an important part of English grammar. In fact, they are included in the common core standards for 8th grade Language, so although most grammarians nowadays will say to go ahead and split the infinitive, students should still understand the concept and decide when to split and when not to split (see what I did there?)
First, a refresher: the infinitive is the word "to" plus the verb. For example, "to love." Splitting the infinitive means to put a modifier between "to" and the verb "love." For example: "to only love" instead of "only to love" or "to love only."
I recently heard a famous forensic linguist (I missed my calling in life!) say that the root of the rule for not splitting infinitives is illogical: In Latin, infinitives cannot be split because they are one word, so it followed that English infinitives should also not be split. He said, then, go ahead and ignore the rule because it makes no sense and we split infinitives all the time and are clearly understood.
He is correct on the second point (the first point - the origin of the rule - is in dispute), but I would say the rule must not be ignored, but understood and then applied or not according to the situation.
Most infinitives can be split and sound perfectly natural; in fact, sometimes it is necessary to avoid awkwardness or changing the meaning of the sentence.
For example, in the sentence "The farmer expects her crop to almost triple next season," the split is necessary because the modifier "almost" can't go anywhere else in the sentence and make sense or not sound awkward.
In addition, take our previous example of the infinitive "to love" and the modifier "only." Moving the word "only" around changes the meaning of the sentence:
Claire wants to only love you.
Claire wants only to love you.
Claire wants to love you only.
The three sentences have different meanings depending on the shade Claire's intentions, and the first requires a split infinitive.
So that leaves the question of whether we should teach students to avoid splitting infinitives when the meaning of the sentence is not affected.
For example:
Split: "The coach expected us to quickly learn the drill."
Not split: "The coach expected us to learn the drill quickly."
I make my students aware of the rule and call it a "best practice," as it sounds better "to the educated ear," as some grammarians put it, but I do not generally correct or mark a split infinitive if it's working well in the sentence. If there is a problem with a split infinitive, it is far easier to explain if they know the grammar and reasoning behind it.
Although you can avoid correcting split infinitives in students' work when it does not affect clarity, it is worth teaching what it is and how to make a conscious choice about its use to avoid a misplaced modifier or an awkward sentence.
In addition, on a purely practical level, if you are teaching for common core assessments on verbals, it is helpful for students who are asked to identify an infinitive to understand that it can be separated with modifiers.