"I am open to become more open-minded about this."
"I am open to becoming more open-minded about this."
Are they both grammatically correct? What's the difference in meaning.
The second sentence (open to becoming) is correct. The phrase "open to [something]" takes a noun or gerund (which is a "nounal verb"): open to suggestions, open to change, open to being corrected.
The first sentence has no real meaning because it is not grammatically correct, though a person hearing the sentence would probably interpret it as it was intended.