All apostles are human, and all humans make mistakes. This doesn't give apostles a license to sin. All believers must guard the heart and the mind and strive to walk before Christ and with each other in integrity. All believers must be on the watch against false motives, wrong attitudes, and choose submission, service, and love. This includes apostles more than anyone.
Do fathers make mistakes? Yes, even the best dads do. This doesn't disqualify them from being a father, or even from being a good father. It certainly does not change a family's need to have such a father, and to continue to receive the love and leadership of such a father. Neither does it remove the love of all the members of the family for the father. Not in a good family. Usually mistakes honestly dealt with will draw a family closer together, and result in increased closeness. In a good family this should happen.
In the family of God mistakes can and will happen. In this way we all learn and we maintain our love and dependence on each other. Mistakes are not a bad thing if they are dealt with properly with a right spirit. This will help bring the whole Church to maturity.
The Scripture has many examples of times when genuine apostles made mistakes, or pursued things which were a dead end, or spoke their own thoughts rather than the Lord's.
"When Peter came to Antioch, I opposed him to his face, because he was clearly in the wrong..." Galatians 2:11
"Forgive me this wrong" - Paul - 2 Corinthians 12:13
The Bible has other examples showing the humanity of apostles, like when Paul in writing Scripture used his own opinions with the words "in my judgement" when talking about marriage. In Acts 15:7-10 Paul tried to cross the border of Mysia and enter Bithynia to preach the gospel, but the Holy Spirit wouldn't allow them. Later that night he had a vision of a man from Macedonia calling them. Paul felt to do a certain thing, but wasn't aware of the Lord's will until it was later revealed. Sometimes apostles (and all believers) will think God wants a certain thing but be mistaken.
Genuine apostles do make mistakes and are fallible, but that doesn't negate the fact that they are called, and have a grace from God for certain tasks. Nobody expects pastors or evangelists or teachers to be infallible, and neither are apostles or prophets. All are called by God, given grace, and anointed and still must walk with Christ in humility to fulfil their call.
Sometimes there are people who claim to be apostles, but have serious error in their lives. For these situations the Bible has clear directions for how to deal with people who are "false apostles" or people who have fallen into serious sin.