it was the mother’s decision
she did not want her around the drugs/sex/rock n roll crowd
1. If she had openly acknowledged Steve Tyler's paternity - rather than lying about it - this would not necessarily have had the consequence that Liv would have been exposed to more drugs/sex/rock n roll than she was anyway. It is very doubtful that Steve Tyler would have insisted upon being awarded custody of Liv, and exposing her to that errant lifestyle.
2. Bebe Buell already had - and persisted in having - a very tumultuous lifestyle, replete with numerous "trysts" with various different rock stars who were heavily into drugs. She was certainly no "choir girl."
3. It is immoral to deceitfully hide such information - unless the father is, e.g., a serial killer.
4. The Wikipedia entries are unclear, so I am unable to confirm this, but Buell apparently also kept the putative father (Todd Rundgren) in the dark, as well. The best I can find is the following quote: "According to Liv Tyler, 'Todd basically decided when I was born that I needed a father so he signed my birth certificate. He knew that there was a chance that I might not be his, but…'"
Regards,