The dot is the problem; it lowers the baseline. I think the easiest way to fix this is to use \smash{...}
to enclose the dotted beta and B on the LHS and the alpha and dotted beta on the RHS. As described in What does \smash do, and where is it documented?, the \smash{...}
command reduces the argument's height and depth to zero, which has the effect of raising the baseline to match that of the other subscripts.
Yes, this this would involve editing each occurrence, but your editor (I use TeXShop and BBEdit on macOS) may have a search/replace feature to automate this for you. I'm sorry that I don't know any other way to do it.
\documentclass[12pt, a4paper]{article}
\usepackage{tensor}
\begin{document}
\begin{equation}
\lbrace \tensor*{S}{^A_\alpha} , \tensor{\bar{S}}{_{\smash{\dot{\beta}B}}} \rbrace = 2 ( \tensor{\sigma}{^\mu} ) \tensor{}{_{\smash{\alpha\dot{\beta}}}} \tensor{K}{_\mu} \tensor{\delta}{^A_B}
\end{equation}
\end{document}
In response to the OP's comments, I searched this site for related issues and here are questions I found:
Differences in vertical positioning of superscripts and subscripts depending on grouping (`{ }`): Why? Caveats? Workarounds? (contains interesting details)
{(x+y)}^2 or (x+y)^2?
How to shift characters and numbers in math mode to make them vertically centered around the formula axis (contains code for visualizing baselines)
Why do all symbols in $x \in X$ have their own baseline? (also contains visualization code)
What is the reason behind the extra vertical space between the superscript and the subscript in a grouping mode? (also includes baseline visualization, but no code for it)
Vertical position of subscripts with commands
It may be worth emailing the tensor
package maintainer to ask if there's a way to address this issue.
UPDATE: The tensor
maintainer has responded and says he will attempt to address the issue.
UPDATE: The tensor
maintainer has released version 2.2 which should address the issue. It should be on CTAN very soon!