In the Ramsey model, how does it make sense to say that capital equals wealth when wealth can be negative and capital cannot? In the Ramsey model, $K_t$ denotes capital, and $A_t$ denotes the assets/wealth of the households. 

However, each household is identical. Since the economy is closed, the assets of each household equal their share of the capital. So, $A_t = K_t/H$, where $H$ is the number of households. 

However, in theory, $A_t$ can be negative. Obviously a household can accumulate debt at some point in their lifetime. But $K_t/H$ can never be negative, since $K_t$ itself, total capital in the economy, cannot be negative. 

So I do not understand how it makes sense in the Ramsey model to talk about "wealth" and "borrowing" and "debt" since $A_t$ is forced to be positive?
No matter in Ramsey model or the basic structure of Solow-Swan model, as long as is in the closed economy. The asset equals to the capital in the equilibrium. 

$A_t$ is the total asset owned by the households, or by the representative household. In the market equilibrium, all the borrowing and lending must cancel within the economy. Therefore, the only asset in positive net supply is capital, which means $A_t = K_t $. The asset equals the capital stock quantity.

Therefore, the asset per person $a_t$ = the capital stock per person $k_t$. It seems like you confuse $A_t$ with $a_t $.