Designing a game requires logical through processes, and an unswerving devotion to making the best game possible, while still keeping enough distance from the project to evaluate it objectively and take criticism well. If you're looking for a design job in conventional gaming, the barrier to entry is fairly low, but so is the pay. You'll need skills in graphic design, layout and statistical modeling.
These skills are similar to what it takes to be a successful electronic game designer, with the addition of having some basic skills with a couple of programming languages, and possibly skills with 3-D rendering and art tools. Being a game designer in an electronic games field means that it's your job to organize how game play is run, it's your job to make sure that the game is engaging and enjoyable, and it's your job to take the feedback of play-testers to heart and improve the game.
One of the places where the two career paths diverge is that the teams that produce board and card games are generally much smaller - so are the companies. They're normally about 3-10 people in size. Development teams for electronic games have deliverable dates that are set two years in advance, and are usually made up of 3-8 game designers, 5 or more artists, and anywhere from 20 to 30 programmers and quality assurance specialists. It's the quality assurance positions (also known as game testers) that give the point of entry into this career field for most people; these are people who play a game again and again trying to find exploits, bugs and things that make the game less fun to play so that they can be solved by the design and coding teams.
It's this large amount of manpower required that makes 'independent' electronic game design such a self limiting field; while it is used to be possible for a single programmer to design, code and generate art for a commercially viable game, that era is at least 20 years in the past, or relies on platforms that are comparatively crude, like cellular phones with limited graphic display systems.
Most entry level game testers make between 18,000 and 24,000 per year. Moving into actual programming and development and design gets into salaried positions of $60,000 per year or more in electronic games. Pen and paper game designers make somewhere between $20,000 and $50,000 per year, but usually live in parts of the country where the cost of living is lower.