Political cartoons and memes are not the best way to communicate arguments (you often need more detail if you want to fully justify a position), but the advantage is that you get quick insight into the way a group of people think and feel. Why are there people actively opposed to voting? The perspective has always struck me as a little odd. These images are provided by a man who posted them in the Mensa forum.
This is the not-my-fault argument. This may be a good analogy at least in part, but it wouldn't help the position, from my perspective. Accidents like this one are caused by too many people driving the wrong way. Fuck-ups in politics are caused by too many voters voting the wrong way. The big difference is that you will be part of the fuck-up regardless.
All the cartoons are wrong in various ways, but this one might be the most wrong. Non-voters and voters alike are involved in and impacted by the accident. They are all drivers on the road, forced to go where it goes and impacted by the rules of the road. Politicians are the road builders who decide where it goes, how many lanes, quality of construction, and the rules of the road. All of these impact both the good outcomes (people get where they want to go) and bad outcomes (accidents). Someone is going to make those decisions, and all drivers are given some input into who those people are. Whether you vote or not, you are responsible for the outcomes. If you choose not to decide, you still have made a choice (Neil Peart, Rush). Non-voters are choosing to leave it up to others to choose who the decision makers are, and that choice ultimately impacts the outcomes. Thus, they are responsible for the outcomes too.
Perhaps a better analogy is that we are on a ship out to sea. That is what existence is and there is no way off it except death. If everyone does nothing, then the ship drifts uncontrolled with certain eventual destruction by running ashore, into an iceberg, or other ships, or a deadly storm. Thus, decisions must be made about how to navigate, where to navigate, what things pose threats, etc.. Everyone cannot make those choices directly, so the passengers each vote to choose the decision makers. Non-voters choose to let everyone else do the choosing. Thus, they are indirectly impacting who the decision makers are by choosing to either to make themselves part of the choosers or not. Those that choose to vote are not any more responsible for the outcomes than the non voters, and may be less responsible. The voters that made an honest effort to make a good choice at least did what they could to reduce the odds of bad outcomes, even if some bad outcomes still occur. The non-voters chose to do nothing to help. They are less culpable than those deliberately making choices that are bad for most people but good for themselves, but more culpable that the people making an honest effort.
The only valid argument for not voting requires one of the following assumptions:
1) All choices are truly identical and have zero impact upon the probability of any possible outcomes.
2) Not voting will somehow cause a fundamental change in the entire system of which decision makers are offerred as options and how they make their decisions, AND, this change will occur soon enough and have enough positive impact that it will offset the damage done by not helping now to make the best choices.
#1 is demonstrably false
#2 has not plausible theory of causality to support it. There is no mechanism by which not voting will force a fundamental change in the system.