Throughout the question, we only consider primes of the form $3k+1$. A reference for cubic reciprocity is Ireland & Rosen's **A Classical Introduction to Modern Number Theory**. 



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> How can I count the relative density of those $p$ (of the form $3k+1$) such that the equation $2=3x^3$ has no solutions modulo $p$? 
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Really, even pointers on how to say anything meaningful about these $p$ are welcome. Originally I also asked about how to count the density of *all* $p$ (not just those of the form $3k+1$) such that $2$ (or $3$) is not a cubic residue modulo $p$, but Felipe Voloch's comment quickly addresses how to deal with them, via Chebotarev's density theorem. 


The difference between the question and these easier problems is that here I am asking that $k+1$ is not a cube modulo the prime $3k+1$, so the same approach does not seem to apply. 



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> Finally, if it turns out that the density is not zero, how does one go about finding the density of those $p=3k+1$ that satisfy that none of the equations $x^3=2$, $x^3=3$, $x^3=k+1$ have solutions?
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(Ideally, the techniques lift to other situations, such as studying fifth powers modulo primes $p=5k+1$, etc, but even methods exclusive to the case of cubes are very welcome.)



Relative densities
A solution can be obtained as suggested by Keith Conrad in the comments, via Chebotarëv's theorem. Details can be found in $\S3.4$ of



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> Coloring the $n$-Smooth Numbers with $n$ Colors
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> Andrés Eduardo Caicedo, Thomas A. C. Chartier, Péter Pál Pach
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> The Electronic Journal of Combinatorics **28 (1)** (2021), #P1.34, 79 pp. DOI: <https://doi.org/10.37236/8492>
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Many thanks to both Felipe Voloch and David E Speyer for their patient explanations.


