Good question. Thanks for asking. The thief on the cross was saved just as Jesus said he was saved. The real question is: can we be saved in the same way that the thief on the cross was saved?
God by His grace has throughout the ages established different covenants with different people. God by His grace gave Noah a covenant to build an ark (Genesis 6:18). Noah was saved because he believed God and obeyed God within the covenant that God gave him.
God by His grace gave a covenant to the patriarchs such as Abraham. Abraham’s covenant was established with him and his seed (Genesis 15:18). Part of the requirements of that covenant were circumcision of all the male children (Genesis 17:10). Abraham was saved because he believed God and obeyed God within the covenant that God gave him.
God gave by His grace a covenant to the Israelites (Exodus 19:5). Part of the covenant that was given to the Israelites was blood sacrifices (Exodus 24:8). The children of Israel were saved because they believed God and obeyed God within the covenant that God gave them.
Today, we are not required by God to build an ark. We are not required by God to be circumcised. We are not required by God to offer blood sacrifices because we are under a New Covenant–the covenant of Jesus Christ (Hebrews 8:7 Hebrews 9:15-17). This covenant was given to us by the grace of God. This covenant came into effect based upon the death of Christ (Hebrews 9:16). The terms of this covenant were given after the resurrection of Jesus Christ (Matthew 28:18,19; Mark 16:15, 16). Under this covenant we are commanded to be baptized (Acts 2:38; Acts 10:48). We believe that God will save us if we believe Him and obey Him within the terms of the covenant that He gives to us.
Under what covenant did the thief on the cross die? He could not have died under the New Covenant because the terms of that covenant had not yet been given when he died. Jesus had not been resurrected from the dead. (The thief died the same day as Jesus.) So he died under the Old Covenant. God saved Him based upon his belief and obedience to that Old Covenant. God will save us based upon our belief and obedience to the New Covenant, so we cannot be saved in the same way that the thief on the cross was saved. Just like we are not saved in the same way that Noah was saved (by building an ark); just like we are not saved in the same way that Abraham was saved (by circumcision); just like we are not saved in the same way that the Israelites were saved (by offering blood sacrifices). We are saved based upon our belief and obedience to the covenant under which God has placed us.
We cannot be saved in the same way that the thief on the cross was saved because we are under a different covenant with God. Baptism is a requirement under the New Covenant and we better be obedient to it if we want God to save us. So under the New Covenant God does require us to be baptized “for the remission of sins.” Please study Acts 2:38; Romans 6:1-10; Galatians 3:28; and 1 Peter 3:21.