The answer depends on your definition of \textit{harmonic}, if you mean $\Delta \alpha=0$ then you can not conclude that $\alpha$ is smooth. It is easy to find counter example on the unit disk in $\mathbb{R}^2$. Indeed a $1$ form on the unit disk, solution of the equation $\Delta \alpha=0$ can be written as  $$\alpha=f(x,y)dx+g(x,y)dy$$ where $f$ and $g$ are harmonic function on the disk, hence $f$ and $g$ are uniquely determined by their boundary value on the circle. Let now $\varphi\colon \partial \mathbb{D}^2\rightarrow \mathbb{R}$ be a continous but not smooth function and let $f$ be the harmonic extension of $y\varphi$ and $g$ be the harmonic extension of $-x\varphi$. Then $\alpha=f(x,y)dx+g(x,y)dy$ solves the equation $\Delta \alpha=0$ and $\left.\iota_{\nu} \alpha\right|_{\partial \mathbb{D}^2}=xf+yg=0$.

If you mean $(d+d^*) \alpha=0$ then it is true because this boundary condition are elliptic (see the chap 5 in the beautiful book of M. Taylor  : Partial Differential Equation I, basic theory).

The answer depends on your definition of harmonic; if you mean $\Delta \alpha=0$ then you can not conclude that $\alpha$ is smooth. It is easy to find counter examples on the unit disk in $\mathbb{R}^2$. Indeed a $1$-form on the unit disk which is a solution of the equation $\Delta \alpha=0$ can be written as  $$\alpha=f(x,y)dx+g(x,y)dy$$ where $f$ and $g$ are harmonic functions on the disk. Hence $f$ and $g$ are uniquely determined by their boundary values on the circle. Let now $\varphi\colon \partial \mathbb{D}^2\rightarrow \mathbb{R}$ be a continous but not smooth function and let $f$ be the harmonic extension of $y\varphi$ and $g$ be the harmonic extension of $-x\varphi$. Then $\alpha=f(x,y)dx+g(x,y)dy$ solves the equation $\Delta \alpha=0$ and $\left.\iota_{\nu} \alpha\right|_{\partial \mathbb{D}^2}=xf+yg=0$.

If you mean $(d+d^*) \alpha=0$ then $\alpha$ is smooth because this boundary condition is elliptic (see the chapter 5 in the beautiful book of M. Taylor: Partial Differential Equations I - Basic Theory).