How should ra'ah be translated in Genesis 18:21?

"‭And the LORD came down to see the city and the tower, which the children of men built.‭"--Genesis 11:5.

The Bible is a masterful system that explains itself. Genesis 11:5 and Genesis 18:21 work together in an explanatory fashion, showing that God does not judge the world by hearsay.

"Not altogether" according to the "cry" of the heavenly record keepers. Rather God will personally "know."

Not that there is some uncertainty, for "‭Neither is there any creature that is not manifest in his sight: but all things are naked and opened unto the eyes of him with whom we have to do.‭" -- Hebrews 4:13.

But rather God seeks to show us all enough evidence that we can intelligently ratify His judgments. He comes to see so that we can see how concerned He is with fairness.

All intelligent beings see enough to trust.

Angels first see the judgment process (Revelation 3:5; Luke 12:8-9). They affirm that God is just. (Revelation 16:3). Then the saints review the judgment process (1 Corinthians 6) even of angels. Finally, the wicked also see the evidence against them and kneel in recognition of God's justice (Jude 1:15; Romans 14:10-12.)

And this is how we understand the word "see" in Genesis 18:21 even if we don't know the first thing about Hebrew grammar or syntax.

The statements of Holy Writ show us a God who invites us to trust Him, not blindly, but with eyes wide open in wonder.

There are a huge number of possibilities but remember we want to translate the most probable. The context of a simple translation "I will now go down and see" gives the sense of the cohortative. We don't need to complicate the translation. The English word see has most of the possible meanings for ראה.

As far as the meanings of see present in Hebrew, Greek, and English, virtually any language:

God will provide for himself the lamb ... (in Gen 22:8, ESV2016)

אֱלֹהִ֞ים יִרְאֶה־לֹּ֥ו הַשֶּׂ֛ה (in Gen. 22:8, BHS2003)

Is more literally saying "God will see to it for himself a lamb..."

Jesus answered them, “Truly, truly, I say to you, you are seeking me, not because you saw signs,... (in John 6:26, ESV2016)

This means that they didn't understand the signs, similar to one saying "I see" meaning "I understand."

Both of these meanings are present in Genesis 18:21, that God will understand and take care of the situation.

ראה:... qal ... —1. to see with one’s eyes: ... —2. רָאָה to see, parallel with יָדַע to understand Is 6:9, to see for oneself,... —3. to see: —a. with לְ of the person ... —4. the object of רָאָה is an independent asyndetic clause: מָה רְאִיתֶם עָשִׂיתִי what you saw me do Ju 9:48. —5. the object of רָאָה is an independent syndetic clause: —a. with כִּי ... —6. a. to spy, reveal Is 28:4 ... —7. רָאָה to see with upset emotions ... —8. to see the face ... —9. a. impv. רְאֵה look! (almost an exclamation) 2S 7:2 —10. רָאָה with acc. of the person, to look at (searchingly) Song 1:6 6:11b. —11. רָאָה to see, meaning to visit, see someone in person, call on —12. רָאָה to choose, select something for oneself,... —13. רָאָה (not with one’s eyes): —a. to understand ... —14. רָאָה with a particular object:...

Koehler, L., Baumgartner, W., Richardson, M. E. J., & Stamm, J. J. (1994–2000). In The Hebrew and Aramaic lexicon of the Old Testament (electronic ed., p. 1157-1159). E.J. Brill.

[Gen. 18:]21. I will go down to see See Comment to 11:5. This stated intention is an element in the motif of theodicy, or the vindication of divine justice, that is essential to the Sodom narrative. As the wickedness of the city appears to reach intolerable proportions, God personally investigates the situation. The fate of the inhabitants of Sodom is not yet sealed. At this point, the humanity of Abraham is put to the test.

Sarna, N. M. (1989). Genesis (p. 132). Jewish Publication Society.