[Dan]

Everytime I balance the CH, the other atoms lose their balance

Sorry, I made a mistake, your goal is 8 Ch on both sides. It is the Ag and CNCh that need looking at in Borek's equation.

[///]

4Ag(s) + 8CNCh(aq) + O2 + 2H2O  >> 4AgCh(CN)2 + 4OHCh

[Borek]

4Ag(s) + 8CNCh(aq) + O2 + 2H2O  >> 4AgCh(CN)2 + 4OHCh

Correct! Now - put negative charge in the place of Ch:

4Ag + 8CN^- + O₂ + 2H₂O -> 4Ag(CN)₂^- + 4OH^-

Check whether the charge is balanced.

Got it?

[AWK]

My addition - balancing under basic conditions means that a base is on the left side of  an equation. When you add water (or nothing) to the left side then we have neutral conditions.

[Borek]

My addition - balancing under basic conditions means that a base is on the left side of  an equation.

So how do you balance half reaction ClO^- -> Cl^- in basic conditions? Looks like you need OH^- on the right, not on the left.

[AWK]

So how do you balance ClO^- -> Cl^- in basic conditions? Looks like you need OH^- on the right, not on the left.

Sorry, but I do not understand your qustion. Redox reactions need reductors and oxidizers.The only reactions I know when CLO- fulfills both reagents is eg
3ClO- = ClO3- + 2Cl-
which proceeds in neutral conditions.
Oxidations by hypochlorite  proceed usually in acidic conditions, ie HClO ia the oxidizing agent.

[Borek]

I suppose you have started answering my post and you have not noticed I have modfied it in the meantime (even if you answered about 12 minutes after I have modified the post). I was referring to the half reaction. Sorry for not being clear.

Oxidations by hypochlorite  proceed usually in acidic conditions, ie HClO ia the oxidizing agent.

I don't think so. Bleach, which is nothing else but hypochlorite solution, is basic (with pH in the range of 12/13) yet is is a strong oxidizer - and to use it you don't have to add any acid. With pKa 7.5 at so high pH almost all hypochlorite is in dissociated form.

[AWK]

I don't think so. Bleach, which is nothing else but hypochlorite solution, is basic (with pH in the range of 12/13) yet is is a strong oxidizer - and to use it you don't have to add any acid. With pKa 7.5 at so high pH almost all hypochlorite is in dissociated form.

Well, though bleach is basic in solution, for classification conditions we use stoichiometry, not actual pH of the reactants, hence oxidation by bleach itself proceeds in neutral conditions. When we add acid, or base to the reaction, conditions are acidic or basic, respectively.
Sometimes reaction is hold up by an  acid or a base formed during reaction, but even then we do not take into account the base or the acid added, since they are not at the left side of chemical equation.

[Borek]

Oxidations by hypochlorite  proceed usually in acidic conditions

oxidation by bleach itself proceeds in neutral conditions

You've lost me here 

[AWK]

Oxidations by hypochlorite  proceed usually in acidic conditions

oxidation by bleach itself proceeds in neutral conditions

You've lost me here 

There is no contradiction in these two statements, oxidation by hypochlorite can be done in neutral or acidic conditions depending on the reaction. I never used a word: exclusively