Romans 13:13 - Let us walk honestly, as in the day; not in rioting and drunkenness, not in chambering and wantonness, not in strife and envying. -- King James Version.
Romans 13: 13 - Let us walk becomingly, as in the day; not in revelling and drunkenness, not in chambering and wantonness, not in strife and jealousy. -- American Standard Version.
The Greek word translated as "honestly" in the King James version is transliterated as euschēmonōs. This is Strong's Greek #2156. Some translations render this word as "becomingly", "decorously", or "decently", or somthing similar. According to HELPS Word-Studies, this word is from Strong's #2158, "eusxēmōn, which is literally 'having good form'... i.e. what is respectable, honorable (modest, noble). The precise nuance of having 'good form' is only determined by the context".
Paul is obviously referring to the Christian's conduct, which would certainly include honesty. But it includes more than just honesty, as reflected by the context. We are to walk as in the day. The "day" could be referring to the coming day, the age to come, as spoken of in Hebrews 6:4,5. The new creature in Christ is counted as already in that time of the new heavens and new earth, as though the present heavens and earth had already passed away. (2 Corinthians 5:17; 1 Peter 3:13; Revelation 21:1-5) Thus, the righteousness of that age should be seen in the conduct of the child of God in this age.