The proposed amendment, similar to a measure that died in the legislature last year, was one of two bail reform proposals unveiled Thursday.
The other would toughen existing bail rules, including revocation of bail for a person arrested for a violent felony while already out on bail.
Biden said the reforms would help protect the public from repeat offenders.
Currently, bail can be denied only for first-degree murder.
Supporters of the constitutional amendment say judges should be able to deny bail for defendants not charged with murder if they believe pretrial detention is the only way to reasonably ensure the safety of the public.