How do I find the degree of a multivariable polynomial automatically?

Perhaps

 Exponent[# /. Thread[Variables[#] -> \[FormalX]], \[FormalX]] &[s1 + s2^2 s3 + s3^2 s4^7]
 (* 9 *)

?

UPDATE: As noted by Daniel in the comments, the original function gives the wrong result if cancellations occur after all variables are replaced by the same symbol. To fix this issue, one can modify the function as

Exponent[# /. ((# -> \[FormalX] RandomReal[]) & /@ Variables[#]), \[FormalX]] & 

(per Daniel's suggestion), or as

Max@Exponent[MonomialList[#] /.Thread[Variables[#] -> \[FormalX]], \[FormalX]]& 

Max[Plus @@@ CoefficientRules[#][[All, 1]]] &@ (s1 + s2^2 s3 + s3^2 s4^7)

=> 9

Edit

J.M. (in a comment) suggests the following neat modification:

Max[Cases[CoefficientRules[s1 + s2^2 s3 + s3^2 s4^7], v_?VectorQ :> Total[v], 2]]

Here's an implementation adapted from a routine by Eric Weisstein:

AlgebraicDegree[eqn_, vars_List] := Max[Total[
      GroebnerBasis`DistributedTermsList[eqn /. Equal :> Subtract, vars][[1, All, 1]], {2}]]

AlgebraicDegree[eqn_] := AlgebraicDegree[eqn, Variables[eqn]]

Example:

AlgebraicDegree[s1 + s2^2 s3 + s3^2 s4^7]
   9