I have 6 different mutation status of a gene. Each group has control samples and drug-treated samples. I want to compare wt vs drug treated in each mutation status.

I can do DESeq for all samples together and then select each mutation status and get results.

dds <- DESeq(dds)
res_KO175 <- results(dds,contrast = c("condition","KO175_DP","KO175_CTRL"),alpha = 0.05)

I can also select each mutation status and do DESeq.

oriKO_oriWT <- DESeq(dds[,c(21,22,17,18)]) # select columns of one mutation status
res_oriKO_oriWT = results(oriKO_oriWT,alpha = 0.05)
summary(res_oriKO_oriWT)

The results are quite different. The latter one gets much less differentially expressed genes.

May I ask which way is more appropriate?

The former pools variance information across all the groups, and thereby increases power to detect differences (due to more degrees of freedom, and likely improved variance estimates). As a general rule, this is the way to go, unless you have one or more groups with quite different within-group variability.