What are the generators of the degenerate affine Hecke algebra $H(k)$ for $k > 0$?
The degenerate affine Hecke algebra $H(k)$ over a field $F$ is isomorphic (as a vector space) to the tensor product $$ H(k)=^{\mathrm{v.s.}} FS_k \otimes F[x_1,\ldots,x_k] $$ where $FS_k$ is the group algebra of the symmetric group generated by simple reflections $s_1,\ldots,s_{k1}$ ($s_i=(i,i+1)$) and $F[x_1,\ldots,x_k]$ is a polynomial algebra. Multiplication is defined so that $FS_k\otimes 1$ and $1\otimes F[x_1,\ldots,x_k]$ are subalgebras, and it is convenient to abuse notation by identifying these subalgebras with $FS_k$ and $F[x_1,\ldots,x_k]$, respectively. Finally, the mixed relations are given by $$ x_{i+1}s_i=s_ix_i +1,\;\;1\leq i\leq k1 $$ and $$ x_js_i=s_ix_j,\;\;j\neq i,i+1. $$ In addition to being a subalgebra, $FS_k$ is a quotient of $\pi:H(k)\twoheadrightarrow FS_k$ obtained by setting $x_1=0$. Note that the first mixed relation can be rewritten as $$ x_{i+1}=s_ix_is_i + s_i $$ so we can write $$x_i=s_{i1}\cdots s_1 x_1 s_1\cdots s_{i1} +L_i$$ where $L_i$ is the $i$th JucysMurphy element. In particular, $\pi(x_i)=L_i$.
I recommend Kleshchev's book `Linear and Projective Representations of Symmetric groups' for further information.
I probably misunderstand the question, but let me give a couple of comments any way.
Generators and relations of degenerate affine Hecke algebra can be found e.g. page 20 section 6 of http://arxiv.org/abs/1008.3739, A categorical approach to classical and quantum SchurWeyl duality Alexei Davydov, Alexander Molev
They have been introduced by Drinfeld to describe SchurWeyl duality for Yangians. (And later by Lusztig for another reasons). There is a whole bunch of SWdualities generalizing classical one to the case of various quantum groups:
Hecke algebra <> U_q(gl) (Jimbo)
Affine Hecke algebra <> Affine U_q(gl)
Degenerate Affine Hecke algebra <> Yangians
Cherednik's double affine Hecke algebra <> toroidal quantum group of type SL(n+1) (M. Varagnolo, E. Vasserot http://arxiv.org/abs/qalg/9506026)
At least the first 3 are compatible in the sense that degenerating the object on the left hand side and on the right hand side we can preserve the duality.
In general I think is amazing and still not clearly understood (if it is possible at all) that Hecke algebra which originally appeared by absolutely different reason, show up themselves in SWduality for quantum groups.
In http://front.math.ucdavis.edu/1009.3295 khovanov gives a nice presentation of the DAHA via a graphical calculus. The generators are then certain diagrams, and computations become somewhat more manageable (at least psychologically)

1$\begingroup$ This is obvious from the question, but just to be clear, the "D" in DAHA here means "degenerate" and not "double". $\endgroup$ – Peter Samuelson May 30 '13 at 20:19