Ugly roots using newtxmath: alternatives

Assuming you want to stick with the newtxtext and newtxmath font packages, I would use the \uproot and \leftroot macros -- provided originally by the amsmath package but included in newtxmath as well -- to fine-tune the position of the nth-root symbol.

There is no single "optimal" value for the arguments of \uproot and \leftroot, as a "good" positioning will depend on the value of the nth root. The following screenshot and code considers the cases 3 and n.

\documentclass[12pt]{article}
\usepackage{newtxtext,newtxmath}
\begin{document}
$\sqrt[3]{2} \quad \sqrt[\uproot{2}3]{2} \quad \sqrt[\leftroot{1}\uproot{2}3]{2}$

$\sqrt[n]{2} \quad \sqrt[\uproot{3}n]{2} \quad \sqrt[\leftroot{1}\uproot{3}n]{2}$
\end{document}

You could keep working with newtxtext and newtxmath, but modify the \sqrt command to move a little the degree of the root.

\documentclass[11pt]{article}
\usepackage{newtxtext}
\usepackage{newtxmath}
\let\oldsqrt\sqrt
\renewcommand{\sqrt}[2][]{%
    \oldsqrt[\raisebox{1pt}{$\scriptscriptstyle #1$}]{#2}%
}
\begin{document}
\begin{tabular}{ll}
    Regular root:               & \( \oldsqrt[3]{2}, \oldsqrt[k]{2}, \oldsqrt[n]{2} \) \\
    With \verb|\raisebox{1pt}|: & \(    \sqrt[3]{2},    \sqrt[k]{2},    \sqrt[n]{2} \)
\end{tabular}
\end{document}

I think it already looks better, especially for the k-th and n-th roots.