I have a few repairs to make that would necessitate some welding. I could do it differently, for example with pop rivets, but I take this opportunity to learn welding. However I'm a bit concerned about the unhealthy part of the process, in particular breathing carcinogen fumes. Is there a way to mitigate those risks ? I'm talking specifically about: Shielded metal arc welding, a.k.a stick welding.
Breathing the vapors is not a good thing. Especially with galvanized metal. It is important to have proper ventilation. Many times when working indoors I use a small fan pointed above the work to move the fumes away and not disturb my shielding gas (wire welding & TIG). When stick welding the fan really has little affect. Breathing the "smoke" caused by welding can cause serious lung problems as it is full of vaporized metal. I work with guys my age that have been professional welders for 30+ years and other than galvanized work they never use a fan or respirator. They laugh at me for my little fan. Welding is a great skill to have and can save you thousands (or if like me you might get bigger and bigger units. On my farm I have repaired all kinds of stuff that would have been recycled if I did not know how to weld. I usually grind the paint so there are less bad things to breathe and use a small 4" fan that has a magnetic base.
Although proper ventilation is probably the most effective defense against breathing metal fumes, I believe that P100 respirator cartridges (the soft pink discs that fit onto a standard half- or full-faced respirator) will remove most of the nasty components of welding fumes, assuming you're willing to wear a respirator while welding.
Engineered controls, such as local ventilation (a unit which removes the fumes from your breathing zone and ejects them away from the area) which can move 100 lineal feet per minute across the weld space, are preferred for metals, rod, and flux that could generate hexavalent chrome fumes / particulates (think cancer) . This would be for any type of stainless steel or alloy containing chrome.
For welding involving carbon steel only, general dilution ventilation is acceptable; basically a fan pulling air from the area, allowing fresh air to enter and dilute the fumes.
A 1/2 face respirator with P100 cartridges should always be worn.
For practical purposes, and as described in the answer by @Joel Keene, the most basic and easily used control would be a respirator with P100 cartridges. Even without ventilation, a respirator would help a lot. A 1/2 face would be what you would want, to facilitate the use of your welding hood or goggles.