Lagrange multiplier sign issue

When a condition $F(x_1,\ldots, x_n)=c$ just defines a set, then this set might as well be defined by the condition $G(x_1,\ldots, x_n)=-c$, where $G:=-F$. Using the first form and the Lagrange function $\Phi:=f-\lambda F$ will produce conditionally stationary points and a $\lambda$-value for each of them. Using the same principle with the second form of the condition, i.e., putting $\Phi:=f-\lambda G$ will produce the same conditionally stationary points, but the associated $\lambda$-values now have the opposite sign. In most examples of multivariate analysis the obtained $\lambda$-value is thrown away anyway; it has no geometric meaning.

In economics things are different. Here a surface $F(x_1,\ldots, x_n)=c$ separates states where some good costs less than $c$ from the states where it costs more than $c$. It follows that the direction of $\nabla F$ carries economical information which should not be thrown away. Therefore it pays to take care of the sign of $\lambda$. I'm not a mathematical economist, whence I cannot tell you what the exact conventions are.

There is no 'right sign', you will just find values for $\lambda$ with opposite signs, but since you're interested in the critical points (to find the min/max), the sign of $\lambda$ isn't relevant. It won't influence the critical points you'll find when solving the system.