I think causal modeling is the key to answering this question. One is faced at the outset to identify the correct adjusted/stratified/controlled effect of interest before even looking at data. If I were to estimate the height / lung capacity relationship in adults, I would adjust for smoking status since smoking stunts growth and influences lung capacity. Confounders are variables which are causally related to the predictor of interest and are associated with the outcome of interest. See Causality from Judea Pearl, 2nd ed. One should specify and power their analysis for the correct confounding variables before the data collection process even begins using rational logic and prior knowledge from previous exploratory studies.
This doesn't mean, however, that some researchers don't rely on data-driven methods to select adjustment variables. I don't agree with doing this in practice when conducting confirmatory analyses. Some common techniques in model selection for multiple adjusted models is forward/backward model selection where you can restrict to classes of models which you believe to be at least plausible. The blackbox AIC selection criteria for this is related to the likelihood and, hence, the degree of reduction in the $R^2$ for linear models for these adjustment variables. Another process common in epidemiology is where variables are only added to the model if they change the estimate of the main effect (like an odds ratio or hazard ratio) by at least 10%. While this is "more" correct than AIC based model selection, I still think there are major caveats in this approach.
My recommendation is prespecify the desired analysis as part of a hypothesis. The age adjusted smoking / cancer risk is a different parameter, and leads to different inference in a controlled study than the crude smoking / cancer risk. Using subject matter knowledge is the best way to select predictors for adjustment in regression analyses, or as stratification, matching, or weighting variables in various other types of "controlled" analyses of experimental and quasiexperimental design.