DRB - the Douay-Rheims Bible - is an English translation published in 1582 of a version of the Latin Vulgate. It is not a translation of the original text of Matthew. Although Matthew may have been written originally in Aramaic1, this edition is lost, so generally the "original" text of Matthew is considered to be the Greek one.
The Greek text here reads:
Ἰδοὺ ἐγὼ ἀποστέλλω ὑμᾶς ὡς πρόβατα ἐν μέσῳ λύκων· γίνεσθε οὖν φρόνιμοι ὡς οἱ ὄφεις καὶ ἀκέραιοι ὡς αἱ περιστεραί.
Aside from punctuation, there is no disagreement between the Textus Receptus (TR)(Scrivener edition), Nestle-Aland (28th ed.) Critical Text (CT), or the 1904 Patriarchal Text (PT) of the Eastern Orthodox Church. The words you ask about are in bold. The 11th edition of the United Bible Societies Greek-English New Testament identifies a couple of minor textual variants, but all contain φρόνιμοι and ἀκέραιοι.
φρόνιμοι (phronimoi) is the plural of φρόνιμος (phronimos), meaning "sensible", "thoughtful", "prudent" or "wise."2
ἀκέραιοι (akeraioi) is the plural of ἀκέραιος (akerios), meaning "pure" or "innocent."3
The Lord warns the Apostle here in advance to be prepared to suffer at the hands of those to whom they will preach:
But beware of men: for they will deliver you up to the councils, and they will scourge you in their synagogues; And ye shall be brought
before governors and kings for my sake, for a testimony against them
and the Gentiles (Matthew 10:17 KJV).
The exhortation to be wise as serpents and harmless as doves is meant to teach them that they must be guileless, but at the same time unfoolish. Theophylact explains here:
[Although they must suffer], He also wants the disciples to be wise.
So that you might not imagine, hearing them referred to as sheep, tha
a Christian must be foolish, Christ says that he must also be wise,
knowing how to act when surrounded by many enemies. For just as the
serpent allows all the rest of its body to be struck but guards its
head, so let the Christian give all of his belongings and even his
body to those who would strike it; but let him guard his Head, which
is Christ and faith in Him. And just as the serpent squeezes through
a narrow hole and sheds its old skin, so too let us traverse the
narrow way [cf. Matthew 7:13] and shed the old man [cf. Ephesians
4:22]. But since a serpent is poisonous, He commands us to be
innocent, that is, sincere, guileless, and harmless as doves. For
when the offspring of doves are taken from them and they are driven
away, they fly back again to their masters. Be wise, then, as the
serpent lest you be tricked in this life, but be blameless in all your
ways; and as for harming others, be as the dove that is
The Douay-Rheims, as mentioned above, is translating from a Latin translation of Matthew's text. The Latin here reads:
Ecce ego mitto vos sicut oves in medio luporum. Estote ergo prudentes
sicut serpentes, et simplices sicut columbæ.
Simplices is the plural of simplex which can mean "simple", but also has the meaning of "honest", "frank", "straightforward" or "guileless."5. the latter meanings here are more consistent with the Greek. Prudentes is the plural of prudens, which means "wise" or "prudent".6
Jerome (347-420) comments in Latin on the Latin text and gives an explanation similar to that of John Chrysostom:
Be simple as a dove and lay snares for no man: but be cunning as a
serpent and let no man lay snares for you. For a Christian who allows
others to deceive him is almost at much at fault as one who tries to
deceive others (Letter LVIII, to Paulinus)
1. Eusebius (263-339), quotes Papias (70-163): "So then Matthew wrote the oracles in the Hebrew language, and every one interpreted them as he was able" (Ecclesiastical History, III.39). In the introduction to his commentary on Matthew, Theophylact of Ohrid (1050-1107) wrote: "Matthew, also known as Levi, tax collector turned apostle, was the first to compose the Gospel of Christ, in Judea in the Hebrew language, for those of the circumcision who believed. it is unknown by whom it was later translated into Greek. The Hebrew text is preserved to this day in the library of Caesarea that was most diligently assembled by the Martyr of Pamphilus. The Nazarenes of Berroia in Syria, who use this text, gave me permission to copy it"
2. F.W. Gingrich and F. Danker, Bauer's Greek-English Lexicon of the New Testament and Other Early Christian Literature (2d ed.; University of Chicago Press, 1979)
4. The Explanation of the Holy Gospel According to St. Matthew (tr. from Greek; Chrysostom Press, 1992), p.85-86
5. J.M. Harden, Dictionary of the Vulgate New Testament (London: Society of Promoting Christian Knowledge, 1921)