The Hebrew text of Gen. 1:26 states,
כו וַיֹּאמֶר אֱלֹהִים נַעֲשֶׂה אָדָם בְּצַלְמֵנוּ כִּדְמוּתֵנוּ וְיִרְדּוּ בִדְגַת הַיָּם וּבְעוֹף הַשָּׁמַיִם וּבַבְּהֵמָה וּבְכָל הָאָרֶץ וּבְכָל הָרֶמֶשׂ הָרֹמֵשׂ עַל הָאָרֶץ
which may be translated as,
26 And God said, "Let us make man in Our image, according to Our likeness, and they shall rule over the fish of the sea, and the fowl of the heaven, and over the beasts, and over all the earth, and over every creeping thing that creeps upon the earth.
However, another possible translation of וְיִרְדּוּ is "so that they may rule." According to this translation, God made man in Their image and according to Their likeness so that they (humans)2 may rule over the rest of Their creation. This would be equivalent to the Greek expression ἵνα ἄρξωσι(ν) expressing purpose (τελικώς).
The Hebrew Verb רָדָה
The Hebrew verb יִרְדּוּ (yir·du') is conjugated in binyan Paʿal from the lemma רָדָה (ra·dah'). In binyan Paʿal, the verb essentially means "to rule," as indicated by its usage elsewhere.3
Heinrich Friedrich Wilhelm Gesenius wrote,4
"In Our Image and According to Our Likeness"
There is essentially no difference in the prepositions ב and כ preceding the nouns in the phrase בְּצַלְמֵנוּ כִּדְמוּתֵנוּ. For, in Gen. 5:1, it states that God created man בִּדְמוּת אֱלֹהִים rather than כִּדְמוּת אֱלֹהִים (despite כִּדְמוּתֵנוּ in Gen. 1:26), and in Gen. 5:3, it states that Adam begat a son בִּדְמוּתוֹ כְּצַלְמוֹ rather than כִּדְמוּתוֹ בְּצַלְמוֹ.
Therefore, the translation "have dominion" is synonymous with "rule." God made man in Their image and according to Their likeness so that man would rule over and subjugate,5 not only the creatures upon the earth and in proximity to it (i.e., those inhabiting the sky and the sea), but the very earth itself.
Notice the following:
When the Lord Jesus Christ calmed the waves of the sea,6 he was demonstrating rule over the earth, the authority which Adam and his descendants were deprived of as a consequence of his transgression.
But the men were amazed, saying, "What kind of man is this, that even the winds and the sea obey him!?
1 The LXX translates the verb וְיִרְדּוּ into Greek as an imperative ἀρχέτωσαν, i.e. "They shall rule!/"Let them rule!"
2 The verb וְיִרְדּוּ is conjugated in the plural number.
3 A master ruling over a slave: cp. Lev. 25:43, 25:46, 25:53; a king ruling over nations: cp. 1 Kings 4:24; Psa. 72:8
4 p. 758
5 cp. Gen. 1:28: וְכִבְשֻׁהָ ("and subjugate it")
6 Matt. 8:27
Gesenius, Heinrich Friedrich Wilhelm. Gesenius’s Hebrew and Chaldee Lexicon to the Old Testament Scriptures. Trans. Tregelles, Samuel Prideaux. London: Bagster, 1860.